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Craftsmanship in the
Teaching of Elementary
Mathematics
BLACKIE & SON LIMITED
50 Old Bailey, LONDON
17 Stanhope Street, GLASGOW
BLACKIE & SON (INDIA) LIMITED
Warwick House, Fort Street, BOMBAY
BLACKIE & SON (CANADA) LIMITED
TORONTO
Craftsmanship in_the_
Teaching of Elementary
^   O  " v
Mathematics
BY
F. W. WESTAWAY
Formerly one of H. M. Inspectors of Secondary Schools
Author of " Scientific Method, its Philosophical Basis and its Modes of A
" Science Teaching: What it Was What it Is What it Might E
" The Endless Quest: 3000 Years of Science " &c.
BLACKIE & SON LIMITED
LONDON AND GLASGOW
First issued 1931
Reprinted 1934, *937
Printed in Great Britain by Blackie & Son, Ltd., Glasgow
" Have some wine" the March Hare said in an
encouraging tone.
Alice looked round the table, but there was nothing
on it but tea. " / don't see any wine" she remarked.
" There isn't any'' said the March Hare.
" Then it wasn't very civil of you to offer it"
said Alice angrily.
" You are sad," the White Knight said. " Let me
sing you a song to comfort you."
" Is it very long?" Alice asked, for she had heard a
good deal of poetry that day.
" It's long" said the Knight, " but it's very, very
beautiful. Everybody that hears me sing it either it
brings the tears into their eyes, or else "
" Or else what?" said Alice, for the Knight had
made a pause.
" Or else it doesn't, you know. The song is called
' WAYS AND MEANS ', but that's only what it's called,
you know!"
" Well, what is the song, then?" said Alice.
" / was coming to that," the Knight said. " The
song really is ' ASITTING ON A GATE ': and the tune's
my own invention"
BY F. W. WESTAWAY
SCIENCE TEACHING
What it Was What it Is
What it Might Be
Second Impression. 10s. 6d. net
" Get the book and read it; it is the best thing yet. It is packed
with practical advice which will always be of value."
Journal of Education.
" His hook wll set many a young teacher on the right path, and
will help many an older one to raise his performance to a much
higher level or excellence." Nature.
"Reveals on every page the zestful interest of a true craftsman in
teaching blended with informed good sense. . . . This book should be
read by all headmasters and headmistresses in secondary schools, and
it is worthy to be studied by every teacher of science. If its counsels
are adopted and followed we shall see a great and beneficent change
in the present method of dealing with science as a factor in education."
Education Outlook.
" This is a remarkable book, critical and stimulating, the product of
the author's long experience as teacher, headmaster, and H.M.I.
. . . comprehensive in scope and so practical that it will be a most
helpful guide to the beginner and an inspiration to all. We recommend
it unreservedly to all engaged in science teaching in schools and
universities." School Science Review.
PREFACE
When asked to write a book on the teaching of Elementary
Mathematics, I felt doubtful as to the avenue by which the
subject might be best approached. During the present cen
tury, the general " policy " and " attitude " to be adopted in
mathematical teaching have been discussed by so many
authorities that there seemed very little new to say. Finally
I decided that classroom craftsmanship might be made a
suitable basis of treatment. Thus the book is not intended for
the experienced teacher who has already acquired skill in his
art, but for the still struggling beginner. In the leading schools,
mathematical craftsmanship probably leaves little to be desired,
but the leaven has yet to work its way into the mass.
From the great variety of topics that come within the
ambit of the various mathematical subjects, I have selected
for treatment those which, in my experience, seem to give
young teachers most difficulty. To treat all topics that come
within the daily practice of mathematical teachers is impossible;
it would mean writing a dozen books rather than one.
I have sometimes been asked if, as an Inspector pursuing
the same daily round year after year, decade after decade, I
am, when listening to lessons in mathematics, ever amused,
ever really interested, ever inclined to be severely critical,
ever bored.
vii
viii PREFACE
Amused? Yes; for instance, when a young master tells his
boys that mathematics is by far the most important subject
they learn, inasmuch as it is the only one that leads them into
the region of " pure thought ".
Really interested? Yes, every day of my life. In the crafts
manship of even a beginner there is almost always some element
of interest; in the craftsmanship of a really skilful mathe
matical teacher there is to me always a veritable joy. I never
enter a classroom without hoping to find something which will
make an appeal, and I am not often disappointed. Sometimes
disappointed, of course; unfortunately not all mathematical
teachers have come down from heaven.
Severely critical? Yes, occasionally, more especially at the
rather slavish adoption of certain doubtful forms of traditional
procedure. For instance, a teacher may include in the work
of the bottom " Set " of a Form the Italian method of division,
well knowing that twothirds of the boys will thenceforth
always get their sums wrong. Another teacher may adopt
" standard form ", not because he has examined it and found
it to be good, but because " everybody does it nowadays ".
Instead of saying, " I thought we had to do these things,"
why do not mathematical teachers hold fast to the faith which
is really in them? If their faith, their faith, includes the Italian
method, standard form, and the score of other doubtful ex
pedients that spread like measles from school to school, I
have nothing more to say.
Bored? Yes, though not often. The petrifying stuff often
doled out to Sixth Form specialists, the everlasting Series and
Progressions, the old dodges and devices and bookwork "proofs"
ad nauseam in preparation for scholarship examinations, all
this is virtually the same now as forty years ago. True, teachers
PREFACE ix
are not much to blame for this. Boys have to be prepared for
the scholarship examinations, and according to prescription.
But that does not soothe an Inspector who has to listen to the
same thing year after year, and I admit that, with Sixth Form
work, sometimes I am almost bored to tears.
If I had to pick out those topics which in the classroom
make the strongest appeal to me, I should include (i) Arith
metic to six, seven, and eightyearolds, when well taught;
(ii) Beginners' geometry; (iii) Upper Fourth and Lower Fifth
work when the rather more advanced topics in algebra, geo
metry, trigonometry, and mechanics are being taken for the first
time (not the Upper Fifth and its revision work); (iv) Upper
Sixth work when examinations are over and the chief mathe
matical master really has a chance to show himself as a master
of his craft. Sixth and Upper Fifth Form work often savours
too much of the examination room to be greatly interesting:
everything is excluded that does not pay. But inasmuch as
examination success is a question of bread and cheese to the
boy, the teacher is really on the horns of a dilemma, and very
naturally he prefers to transfix himself on that horn that brings
him the less pain.
Why, of all the subjects taught, is mathematics the least
popular in girls' schools? and whyjs [t_the one subject in
which the man in the street feels no personal interest?
It is not because mathematics is difficult to teach. My own
opinion is that it improbably the easiest of all subjects to teach.
When it is taught by wellqualified mathematicians, and when
those mathematicians are skilled in their teaching craft, suc
cess seems always to follow as a matter of course, in girls'
schools equally with boys'. The failure to make any head
way, even under the best conditions, on the part of a small
x PREFACE
proportion of boys and a rather larger proportion of girls is
probably due to a natural incapacity for the subject. Had
I my own way, I would debar any teacher from teaching even
elementary mathematics who had not taken a strong. doge^ of
the calculus, and covered a fairly extensive field of advanced
work generally. It is idle to expect a mathematical teacher to
handle even elementary mathematics properly unless he has
begjx.through the mathematical mill. And yet I have heard
a Headmaster say, " He can take the Lower Form mathe
matics all right; he is one of my useful men: he took a Third
in History."
As long as University Scholarships are what they are, so
long will Sixth Form specialists* work proceed on present
lines. But one purpose of the book is to plead for considera
tion of the many neglected byways in mathematics and for
their inclusion in a course for all Sixth Form boys; suggestions
to this end are made in some of the later chapters. We want
a far greater number of ordinary pupils to become mathe
matically interested, interested in such a way that the interest
will be permanent; and we want them to learn to think mathe
matically, if only in a very moderate degree. Why do ordinary
pupils shrivel up when they find a mathematician in their
midst? It is simply that they are afraid of his cold logic.
There is, in fact, a curious popular prejudice against mathe
maticians as a class. It probably arises from the fact that we
*~~_.._ .~  ., * j
are not a nation of clear thinkers, and we dislike the few amongst
us who are. Foreigners at least the French, the Germans,
and the Italians are mathematically much keener than we
are. They seem to become immediately interested in a
topic with mathematical associations, whereas we turn away
from it, disinclined to take part in a discussion demanding
PREFACE xi
rigorous logical reasoning. Competent observers agree that
this is in no small measure traceable to the fact that our
school mathematics has not been of a type to leave on the
minds of ordinary pupils impressions of permanent interest.
We^ have driven Euclid out of Britain, but we must all
admit that he stood as a model of honest thinking, and we
miss him sadly. Were he to come back, frankly admitting his
failings and promising reform, not a few mathematicians
would give him a warm welcome. It is only a very few years
since I heard my last lesson in Euclid, and that, curiously
enough, was at a Preparatory School. It was a pleasure to
hear those 12yearold boys promptly naming their authority
(e.g. I, 32; III, 21; II, 11) for every statement they made, and it
was exceedingly difficult to improvise the necessary frown of
disapproval. Of course those small boys did not understand
much of what they were doing, and of actual geometry they
knew little. But in spite of this they were learning to think
logically, and to produce good authority for every assertion
they made. Our modern ways are doubtless better than the
old ways, but when we emptied the bath, why did we throw
out the baby?
The very last thing I desire to do is to impose on teachers
my ideas of methods. Anything of the nature of a standardized
mgthpd in English schools is unthinkable. The Board of
Education, as I knew it, never issued decrees in matters affect
ing the faith and doctrines of our educational system; it con
fined itself to making suggestions. Admittedly, however,
democracy has now come to stay, and its unfortunate though
inevitable tendency to standardize everything it controls may
ultimately prove disastrous to all originality^in teaching methods,
and reduce the past high average ofjnitm^ve^agdjrf
xii PREFACE
independence in schools. Let^every; teacher^strive to base his
methods., on a venturous originality. Let him resist to the
death all attempts of all bureaucrats to loosen the bonds of
obligation to his art, or to mar his craftsmanship.
But though I plead for originality I desire to utter a warn
ing against a too ready acceptance of any new system or method
that comes along, especially if it is astutely advertised. It is
perhaps one of our national weaknesses to swallow a nostrum
too readily, whether it be a new patent medicine or a new method
of teaching. What good reason have we for thinking that a
teacher of 1931 is a more effective teacher than one of 1881, or
for that matter of 2000 years ago? What is there in method, or
in personal intelligence, that can give us any claim to be better
teachers, better teachers, than were our forefathers? When
a new method is announced, especially if it be announced with
trumpets and shawms, write to the nearest Professor of
Education, and more likely than not he will be able to give
you the exact position of the old tomb which has been recently
ransacked.
A method which is outlined in a lecture or in a book is
only the shadow of its real self. A method is not a piece of
statuary, finished ~and unalterable, but is an_^er^chajniging
thing, varying with the genius of the particular teacher who
handles it. It is doubtful wisdom to try to draw a sharp
antithesis between good methods and bad: the relative values
of abstractions are invariably difficult to assess. The true
antithesis is between eff^tiy^jfli^^^ The
method itself counts for something, but what counts for very
much more is the life that the craftsman whenjictually at work
breathes into it.
The regular working of mathematical exercises is essential,
PREFACE xiii
for the sake not only of the examination day that looms ahead
but also for illuminating ideas and impressing these on the
mind. Nevertheless, the working of exercises tends to dominate
ourwork far too much, and to consume tiifte that might far
more profitably be devoted both to the tilling of now neglected
ground of great interest and to the serious teaching of the
most fundamental of all mathematical notions, namely, those of
number, function, duality, continuity, homogeneity, periodicity,
limits, and so forth. If boys leave school without a clear grasp
of such fundamental notions, can we claim that their mathe
matical training has been more than a thing of shreds and
patches? I plead for a more adequate treatment of these things.
The terms " Forms " and " Sets " I have used in accord
ance with their current meaning. The average age of each
of the various sections of Form II, III, IV, and V is considered
to be 12+, 13+, 14+, and 15+, respectively, the units figure
of the age representing the Form: this is sufficiently accurate
for all practical purposes. The age range of Form VI is taken
to be 16+ to 18+. " Sets " represent the redistributed mathe
matical groups within any particular Form; for instance,
100 boys in the different sections of Form IV might be re
distributed into 4 Sets, a, /?, y, and S. Admittedly it is in the
lower Sets where skilled craftsmanship is most necessary.
On reading through the manuscript I find that I have
sometimes addressed the teacher, sometimes the boy, rather
colloquially and without much discrimination. I crave the
indulgence of my readers accordingly.
Ait teachers of mathematics should belong to the Mathe
matical Association. They will then be able to fraternize
xiv PREFACE
periodically with the bestknown and most successful of their
fellow workers. The Mathematical Gazette will provide them
with a constant succession of lucidly written practical articles,
of hints and tips, written by teachers known, by reputation if
not personally, to everybody really interested in mathematical
education; also with authoritative reviews of new mathematical
books. Members may borrow books from the Association's
Library, and the help and advice of specialists are always to
be had for the asking.
In writing the book my own views on numerous points
have been checked by constant reference to Professor Sir
Percy Nunn's Teaching of Algebra and its two companion
volumes of Exercises, books I have not hesitated to consult
and to quote from, in several chapters. I am old enough to
remember the great reputation Professor Nunn enjoyed as a
gifted teacher of mathematics when he was an assistant master
thirty years ago. The methods he advocates are methods which
have been amply tested and found to be sound and practical.
His book deals with algebra in the broadest sense and gives
advice on the teaching of trigonometry, the calculus, and
numerous other subjects.
F. W. W.
CONTENTS
CHAP,
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
IX.
X.
XL
XII.
XIII.
XIV.
XV.
XVI.
XVII.
XVIII.
XIX.
XX.
XXI.
XXII.
XXIII.
XXIV.
XXV.
XXVI.
TEACHERS AND METHODS 
WHICH METHOD: THIS OR THAT? 
" SUGGESTIONS TO TEACHERS "
ARITHMETIC: THE FIRST FOUR RULES  ...
ARITHMETIC: MONEY 
ARITHMETIC: WEIGHTS AND MEASURES 
ARITHMETIC: FACTORS AND MULTIPLES 
ARITHMETIC: SIGNS, SYMBOLS, BRACKETS. FIRST NOTIONS
OF EQUATIONS
ARITHMETIC: VULGAR FRACTIONS     
ARITHMETIC: DECIMAL FRACTIONS  ...
POWERS AND ROOTS. THE A B C OF LOG
RATIO AND PROPORTION 
COMMERCIAL ARITHMETIC
ARITHMETIC:
ARITHMS
ARITHMETIC:
ARITHMETIC:
MENSURATION 
THE BEGINNINGS OP ALGEBRA
ALGEBRA: EARLY LINKS WITH ARITHMETIC AND GEOMETRY
GRAPHS  
ALGEBRAIC MANIPULATION 
ALGEBRAIC EQUATIONS
ELEMENTARY GEOMETRY 
SOLID GEOMETRY ...
ORTHOGRAPHIC PROJECTION
RADIAL PROJECTION .......
MORE ADVANCED GEOMETRY ....
GEOMETRICAL RIDERS AND THEIR ANALYSIS   
PLANE TRIGONOMETRY ......
Page
1
16
20
23
42,
47
54
62
67
78
94
100
109
118
122
132
137
177
205
225
287
293
300
308
319
XVI
CONTENTS
CHAP. Page
XXVII. SPHERICAL TRIGONOMETRY      381
XXVIII. TOWARDS DE MOIVRE. IMAGINARIES  387
XXIX. TOWARDS THE CALCULUS      401
XXX. THE CALCULUS. SOME FUNDAMENTALS    421
XXXI. WAVE MOTION. HARMONIC ANALYSIS. TOWARDS
FOURIER  451
XXXII. MECHANICS 480
XXXIII. ASTRONOMY 497
XXXIV. GEOMETRICAL OPTICS 506
XXXV. MAP PROJECTION 517
XXXVI. STATISTICS 638
XXXVII. SIXTH FORM WORK 653
XXXVIII. HARMONIC MOTION 558
XXXIX. THE POLYHEDRA 672
XL. MATHEMATICS IN BIOLOGY 584
XLI. PROPORTION AND SYMMETRY IN ART    688
XLII. NUMBERS: THEIR UNEXPECTED RELATIONS   694
XLIII. TIME AND THE CALENDAR 614
XLIV. MATHEMATICAL RECREATIONS 615
XLV. NONEUCLIDEAN GEOMETRY     617
XLVI. THE PHILOSOPHY OF MATHEMATICS    623
XLVII. NATIVE GENIUS AND TRAINED CAPACITY  639
XLVIII. THE GREAT MATHEMATICIANS OF HISTORY   643
XLIX. MATHEMATICS FOR GIRLS 644
L. THE SCHOOL MATHEMATICAL LIBRARY AND EQUIP
MENT 646
APPENDICES
I. A QUESTIONNAIRE FOR YOUNG MATHEMATICAL
TEACHERS 663
II. NOTE ON AXES NOTATION 657
INDEX 659
PLATE
POLYHEDRAL MODELS
Facing
Page
 584
CHAPTER I
Teachers and Methods
Mathematical Knowledge
" That man is one of the finest mathematicians I have
ever had on my Staff, but as a teacher he is no good at all."
" Oh yes, he can teach all right. He can hold a class of 40
boys in perfect order for an hour. The trouble is that his
knowledge of mathematics is so superficial."
These criticisms of Head Masters are not altogether un
common. A Head Master naturally looks for two things in
members of his Staff: sound knowledge of a subject, and skill
in teaching that subject.
Suppose that a teacher has spent three or four years at
the University, has obtained the coveted First in mathematics,
and has then spent a year in the Training College Department
of a University or of a University College. Can he then claim
to be a competent mathematician and a skilful teacher? The
answer is no.
The knowledge of mathematics obtained in a four years'
University course forms an admirable foundation on which to
build, but how much mathematics can be learnt in so short a
period as four years? At the end of that time it is a common
thing for University students hardly to have touched the serious
mathematics of physics, of chemistry, of engineering, of ma
chinery and structures, of aviation, of astronomy, of biology,
of statistics, or to have mastered more than the barest elements
of the philosophy of the subject. Assuming that it takes a year
to acquire an elementary knowledge of each of the applied
(E291) 1 2
2 CRAFTSMANSHIP IN MATHEMATICS
subjects just named, the newlyfledged graduate has still in
front of him a long spell of hard work before he can claim to
be a mathematician in the broader sense of the term. As for
the philosophy of mathematics, he might still be a babe in
the subject after five years' hard reading. Mathematics touches
life at so many points that the allround training of a mathe
matician is a very long business. If by the age of 35 a mathe
matician has acquired a fair general knowledge of his subject,
he has done well. Consider the mathematics of physics alone:
what a vast field! The field is, of course, ploughed up and
sown by the teachers of physics, with the result that there
is always a rich harvest for the mathematical staff to reap.
Is that reaping always done?
Skill in Teaching: Training
Then as to teaching. How can a mathematical Training
College student claim to be an efficient teacher at the end
of his training year? Skill in any art can be acquired only by
much practice, and the art of teaching is_a particularly diffi
cult art. Although mathematics is one of th^BaQioot, perhaps
the easiest, of all subjects to teach, it is a rare thing for a
mathematical teacher to be able to feel at all satisfied with
his professional skill before the age of .30. He is lucky if
other people adjudge him efficient before the age of 35. Every
mathematical subject is full of teaching problems. Every
one of these problems can be solved in a variety of ways.
Every one of these ways is worth testing. And all this takes a
long time. As for lucidity of presentation a prime necessity
in all mathematical teaching that is in itself an art which
half a lifetime does not seem long enough to perfect.
I have often been asked by mathematical: teachers who have
not been formally trained what good they would have done
by going to a Training College. They are inclined to argue
that very few of the Training Colleges have on their Staffs
a frontrank mathematician, and that therefore such Training
Colleges are not in a position to deal with the subject effec
TEACHERS AND METHODS 3
tively; that even if the Training Colleges are able to impress
into their service members of the mathematical staff of the
local University, little real help is obtainable, for " although
a University Professor can teach me mathematics, he cannot
teach me how to teach boys mathematics ". The general con
tention is not without its points, but the strictures certainly
do not apply to all Training Colleges, as trained mathematical
teachers who have been through the hands of, say, Professor
Sir Percy Nunn are the first to admit.
The further criticism that the time spent on Psychology at
the Training Colleges is " absolutely wasted ", since " it has
no practical value in the solution of actual classroom prob
lems ", is, perhaps, rather more justified. It is possibly true
that the almost useless introspective psychology of half a
century ago still hangs about some of the Training Colleges;
I do not know. But young teachers should make themselves
acquainted with the valuable experimental work which is now
being done by psychologists all over the world. These ex
periments are often based on masses of actual data derived
from the classroom. It is true that the definite results obtained
so far are rather patchy; a complete body of psychological
doctrine has yet to be built up into something that may claim
to be " science ". But no teacher can afford to ignore the
work that has been done and is being done, if only because
such a large part of it has a very close bearing on presentday
school practice.
One broad distinction between the outlook of a teachei
who has been through a Training College and that of one
who has not is that the trained teacher has usually had knocked
QUL.Qf_hini_the jprejudice which he (very naturally) felt for
his own special subject, whether mathematics, classics, or
what not. He has learnt that in the Common Room he will
become a^ member of a community regresenting^alj the subjects
of the curriculum, and that " all these subjects, not : Jiis own
subject^alone, answer to deeseated neejs_ofjhe human
spirit, all of tftern es^ in the great stream of
movement called^ civilization ". The Training College does
4 CRAFTSMANSHIP IN MATHEMATICS
its best, of course, to turn out competent craftsmen, but it
does much more. It leads its students to understand the
real meaning of education and something of its significance
in relation to the manysided business of life. It shows them
how much wider education is than mere teaching. Moreover,
the students are day by day in contact with men who have
reflected deeply upon both education in the broader sense
and teaching in the technical sense.
In a good Training College, the embryo mathematical
teacher is taught not only how to convey to his pupils a know
ledge of arithmetic, algebra, geometry, and the rest, but also
how to make himself, in the greatest measure possible, an
active intellectual adventurer in the realms of number and
space, how to follow up the labours of the great masters of
mathematical thought, and how to catch something of their
spirit and outlook. He is encouraged to question accepted
mathematical values, and to inquire, in a critical spirit, what
parts of the traditional curriculum are really vital and what
parts have only a conventional value. He is made aware that
many of the textbooks contain a considerable amount of useless
lumber, and he is taught: how to discriminate between methods
that are sound and methods that are otherwise.
Then again, in a good Training College the student is
able to^obtain expert advice on every kind of difficulty he may
meet with in his teaching practice. At what stage, for example,
should " intuition " work in geometry give way to rigorous
proof? How can the^best approach be made in the teaching
of ratio and proportion? "How is the theory of parallels to
be treated? The Training College may have to tell him frankly
that in his present state of pupilage he is probably not yet
fitted to deal with the theory of parallels except in an empirical
way, since the inherent difficulties of the theory can only be
grappled with after a prolonged and careful study of the re
searches of modern geometry; only then will he be in a position
to disentangle logic from intuition, and so be able to devise a
treatment suitable for Sixth Form boys. Briefly, the Training
College will point out quicksands, as well as firm rock, for in
TEACHERS AND METHODS 6
mathematical teaching quicksands abound. A teacher who is
not trained will often not only get into the quicksands but take
his boys with him.
It may be said that the great majority of Secondary School
teachers, especially teachers in the Conference * schools, are
not trained. That is true, and the result has often been that
such teachers have bought their first few years* experience at
the ^xpense of their boys. But not always. If a young man
straight down from the University goes as a Student Teacher,
that is as an observer and learner, in a large school where
mathematics is known to be well taught, and if the Head of
the mathematical department has sufficient leisure to guide
him in his reading and to act as his professional friend, tjhtf
year's training may not be inferior to that at a Training College,
Some authorities urge that it may be superior, inasmuch as
the Student Teacher spends the year with real practitioners
rather than with theoreticians. I do not attempt to decide
this question.
Still another alternative a very common one is for the
embryo teacher, just down from the University, to join the
Staff of a big school, to admit freely that he is a neophyte, and
to beg for all the help and criticism he can obtain from his
mathematical colleagues. Let him invite his seniors to come
to hear him teach, and to criticize him, and let him beg the
privilege of being present occasionally at their lessons. Let
him seek their advice as to a suitable course of reading. But
let him not think that he is a teacher sent straight from heaven
to rectify the views and methods of the old fossils of 25 and
upwards he may find in the Common Room: that way lies
trouble.
One thing that the untrained beginner should never do
is to join the Staff of a small school where he has to undertake
the respcfnsfbffify of tR^whele of the mathematical teaching.
In the first place it is not fair to the school; the boys are sent
there to be taught. In the second place it is not fair to himself,
*The term "Public" school is now ambiguous, and is better not used. All
grades of schools are either "public" or "private".
6 CRAFTSMANSHIP IN MATHEMATICS
for how is he to learn his job? Why not open an office, and
set up as a Consulting Civil Engineer instead? He is just as
competent to do the one thing as the other.
In the old days, all wouldbe craftsmen joiners, brick
layers, mechanics, and others served a seven years' apprentice
ship, and they really learned their business; they became skilled
craftsmen. The system is dead, their work being done mostly
by machinery. But we cannot teach by machinery yet; and
skilled craftsmanship in teaching can be acquired only by a
great deal of practice.
Conventional Practice
How is the value of a lesson in mathematics to be assessed?
I do not refer to the ordinary things in which every teacher
with a year's experience ought to be reaching efficiency
classmanagement, discipline, use of the blackboard, expertness
in questioning and in dealing with answers, and so forth
but to the lesson as a mathematical lesson. The commonest
fault of the young mathematical teacher is that he talks too
much; he lectures, and, if he is teaching the Sixth Form, he
often uses his University notes. It takes some young teachers
a long time to learn the great lesson that the Jthing that matters
jft.QSLJg. not what Jthey give out but what the boys take in;
that their work is teaching, not preaching."""""
Another common fault of young mathematical teachers,
and not all experienced mathematical teachers are guiltless of
it, is the adoption of a particular method because it is mathe
matically neat, the sort of method that appeals to a mathema
tician as a mathematician, not the method that is the most
suitable for demonstrating a particular principle or teaching
a particular rule so that the child can understand it._ If a
mathematical teacher thinks that a mathematician is listening
to him, he is more often than not keener to reveal his know
ledge of mathematics than to exhibit his teaching power
But the observer's object is usually to discover what the boys
are learning, and to assess the value of the teaching; and the
TEACHERS AND METHODS 7
very u neatness " of the method adopted is quite likely to be
the cause of the boys learning next to nothing. To be effective,
a method must be simple and be clear to the boys. Mathe
matical rigour may thus have to Be""sacrlftced, though the
rigour then sacrificed will come later.
Again, young mathematical teachers are ajrt to be hide
bound b^coiwe^ . Generation after generation of boys
are told, for example, that, in an expression like m + n X p,
the multiplication sign should take precedence over the addition
sign. Why should it? Doubtless the original suggestion is
hidden away in some old textbook, but it has been consistently
adopted by modern writers as if it were something sacrosanct.
Surely if the signs are not to be taken in their natural or3er
from left to right, it is the business of the person who frames
the question to insert the necessary brackets, and not leave
the wretched little learner to do it.
Let the Head of a mathematical department in a big school
remember that the place of honour for himself or for any
exceptionally gifted member of his Staff isjn the lower Forms.
The beginners' geometry is, more than any other mathematical
work, in need of skilful teaching. The hackneyed stuff usually
doneT by the Sixth Form specialists can quite well be taken
by a youngster just down from the University. He may not
be able to teach, but he is mathematically fresh, and, if the
specialists in the Sixth have been previously well trained,
they can usually take in the new mathematics even if it is
rather clumsily presented to them.
The textbooks mentioned in the course of this volume
are intende^Tm the main, foLJhose teachers to read, who are
technically untrained. The object is not to recommend this
'Book "or that took for adoption in schools: that is not part oi
our purpose at all. The object is to suggest a book, written
by some skilful teacher, for the novice to read right through,
critically. He should ask himself why the writer has approached
and developed the subject in that particular way. He should
then read a second book, then a third, and so on, noting the
different ways of approach and of development, and the different
8 CRAFTSMANSHIP IN MATHEMATICS
ways in which different teachers do things. Then he should
settle down and evolve methods for himself. He should not,
unless in exceptional circumstances, copy another teacher's
method. Let his methods be part of himself, things of his
owiT elation, things for which he has an affection because
they are his own children.
Let him realize that methods of teaching mathematics, as
of teaching other subjects, are largely conventional. What is a
" best " method, and how is it to be determined? Is it a specially
" neat " method, invented by some clever mathematician? If
so, is it a simple method? Is it productive of accuracy? Here,
psychology teaches us a little, though not yet very much,
and to say that one method is " better " than another is, more
often than not, merely to express a personal preference. The
teacher should always ask himself, which method works out
best in practice! Let every teacher make up his own mind,
and not be led away either by the textbooks or by the critics,
though the textbooks will always help, and the critics, if
competent, are worth listening to. But, however good the
books and however competent the critics, let him take their
help and advice critically.
Psychology kas k?lE<Lus great deal over certain points in
the teaching^ oT arithmetic. Experiments have been directed
marnly'to discovering which of possible alternative methods
is productive of greatest accuracy amongst children. If such
experiments are sufficiently numerous and varied, and if the
results of the tests are fairly uniform, we may feel it advisable
to consider a particular method favourably. But people who
experiment in this way must set out with an entirely unpre
judiced mind. Results that are not arrived at objectively
carry no weight.
Whether psychology has yet succeeded in devising con
vincing testsjpf personal matb^atkal^ ability, I am un
certain. The validity of some of the criteria used has been
seriously questioned by recognized authorities. Thejrelatipns
between mathematical ability and general " intelligeace "
have certainly not been clearly determined. We have probably
TEACHERS AND METHODS 9
all met highly intelligent men with keen logical powers who
were no good at all at mathematics, and have known brilliant
mathematicians whose lack of general intelligence in non
mathematical affairs was amazing. We do not yet really
know if mathematical ability can be trained, or whether it is,
so to speak, a fixed quantity at birth. The deductions we can
legitimately draw from mere examination successes are by
no means certain; even poor mathematicians may become
adepts in the use of crammers' dodges.
The "Dalton Plan "
The question is sometimes asked, what is the Dalton
plan of teaching, and can it be made to apply to mathematics?
The plan originated in America ia 1920 and has since
been introduced into a certain number of English schools.
" The aim is to provide for the differences encountered in
individual pupils." Class teaching as such is abolished, and
gives way to organized private study, ijn^wWch^jhe_Uil,
not the teacher, becomes the principal and responsible agent.
Instead oFT course of lessons prescribed according to a
timetable, an " assignment " of work, to last for a month,
is prepared by the teacher. The whole " plan " hinges on
these assignments. The month's task is divided into four
weekly allotments, which are further subdivided jnto daily
units. Instead of working to a timetable, the pupil is free
to work at whatever subject he pleases. The rooms are no
longer " classrooms ", but subject rooms, each being in
charge of a specialist teacher and being provided with the
necessary books and material. The pupils move freely about
from room to room. The instructors are consulted at any
time by any pupil; it is their duty to advise and help when
ever asked to do so. Conferences and collective discussions
are, Jipwever, arranged at specified Tipurs.
""""' There is a certain amount of acceptable opinion in favour
of the plan as regards subjects like English and History, but
as regards Mathematics, Science, and Modern Languages,
10 CRAFTSMANSHIP IN MATHEMATICS
the balance of opinion is undoubtedly against it. For one
thing, theTmajority of mathematical textbooks are unsuitable;
they do not demonstrate and elucidate principles simply
enough for average pupils to understand, with the conse
quence that, in some schools which are working on Dalton
lines, formal lessons on new principles precede the work by
assignments, which, for mathematics, are not much more
than a few general directions, and exercises to be worked.
In short, the plan does not at present seem to be favoured
very much by the majority of teachers. A teacher who adopts
the plan, no matter what his subject, must be prepared for
greatly increased personal labour; if his subject is mathe
matics, he must be prepared for some measure of disappoint
ment too. On the other hand, the able mathematical boy, if
given a free hand, with just occasional help when difficulties
are serious, seems to run away quickly from all the others.
The plan seems to pay with Sixth Form specialists, who have
been^weil trained up to the Fifth. Such boys, if provided
with good textbooks, can, with very little formal teaching or
other help, make remarkably rapid progress.
The one general conclusion that seems to emerge from
Dalton experiments is that pupils would do better if left to
wrestle more for themselves, anckthat in the past we have all
tended to teach too much. Although thejplan as ajglan is, in
the estimation of not a few good judges, rather_top revolu
tionary for gejrier_aJLadpptipn, it must, on the other hand, be
admitted that a clever teacher who loves teaching for its own
sake may be something of a danger; hejrnay do too much
of the thinking, and leave the boys too little to do for them

No boy can become a successful mathematician unless
he rights hard battles on his own behalf.
TEACHERS AND METHODS 11
Some General Principles
The last statement does not mean that mathematical
teaching is not necessary. For all pupils save perhaps the
very best, it is fundamentally necessary, and above all things
the teaching must be clear. Strive day by day to make the
expression of your meaning ever clearer. Choose your words
carefully and use them consistently. Never mind the correct
formal definitions of difficult terms. Use a term over and
over again always in exactly the same sense but associated
in different ways with different examples, until its exact
significance imposes itself on the pupil's mind. It is merely
a question of the child continuing to learn new words much
in the same way as he learnt the stock of common words
which are already in his possession. His mother did not
define for him as a baby such words as milk, mamma, toes,
pussie, sleep, naughty, yet he learnt to understand their
meaning almost before he could walk.
In short, dojiot worry beginners with formal definitions,
or abstractions of any other sort. Of course, almost from
the first, the boy makes crude use of ajl sorts Q.f crudely
acquired abstract terms, for in his enumeration work and in
his early quantitative measurements, which he has always
associated with concrete objects, intuition and guesses have
played a large part. But the mathematical ideas and pro
cesses which he uses for solving different practical problems
gradually become clearer, and he begins to see interrelations
between principle and principle, and to distinguish those
which are mutually connected from those which are inde
pendent. As the subject proceeds, it tends to become more
abstract; experience grows; and the teacher has to choose
his own time for stepping in and exacting greater and greater
logical rigour. Below the Sixth Form, mathematics is essen
tially a practical ^^ instrument, not a subject for philosophic
speculation. Never press forward formal abstract considera
tions until ' ~~ "*
12 CRAFTSMANSHIP IN MATHEMATICS
What is the use of discussing even with Fifth Form pupils
the rival merits of Euclid's parallel postulate and Playf air's
alternative version? For all lower and middle forms, some
such statement as, " lines which intersect have different
directions; lines which have the same direction do not meet
but are parallel ", is good enough, and it need not be sub
jected to criticism until the Sixth. Then, criticism is desirable.
The organization of mathematical work in a large school
is a simple matter; between the Junior Forms and the Sixth
there may be 4 blocks of 3 or 4 Sets each. When the Set
system prevails, gradation is easy. Let the work of the top
Set of a block be much sterner and more exacting than in
the bottom Set, and do not attempt to include in the work of
the bottom Set all the subjects, or even all the topics of a
particular subject, that are allotted to the better Sets. For
instance, all top Sets will learn logarithms. But bottom Sets?
Why should they? What difficult calculations will they have
to engage in that logarithms will really help? None in school,
and none after leaving school. Why then should such dull
boys be made to waste their time by poring over the pages
of a numerical lexicon and then getting their sums wrong
instead of right? It is unutterably silly. It is sometimes done
because teachers have not the courage to say what they really
think.
The timid teacher may be inclined to argue, " but how are
we to provide for the boy who during the year happens to be
promoted a Set?" That is certainly a real problem of school
organization and must be faced. But the needs of the occa
sional boy must not be catered for at the expense of a whole
class. And, after all, there will be much the same minimum
of work for the various Sets within a block, and interblock
promotions after the first year or two will be rare.
Again, suppose that somebody comes along and asks if
you teach, say, Vectors. If you do not, you probably have
a good reason for it, perhaps because Lord Kelvin himself
poured scorn on them. In that case do not hesitate to say so.
Holdfast to your faith. But reexamine the grounds of your
TEACHERS AND METHODS 13
faith from time to time. It may be that you will find new
arguments in favour of vectors, arguments which will induce
you to revise your opinions. And so with scores of other
things. Keep an eye on your defences, but remain captain
of your own quarterdeck.
Mathematical Reasoning
All mathematical teachers should reflect carefully on the
nature of mathematical reasoning, and should see that their
pupils are made more and more conscious of what constitutes
mathematical rigour. Mathematical reasoning is not, as com
monly supposed, deductive reasoning; it is based upon an
initial analysis jrf jthjy^en. and, being analytical, is,jn essence
' ~* " ' e. The threads of the web once disentangled, synthesis
begins, and the solution of the problem is set out in deductive
4yqgs. We arrange our^ arguments ded^
y easily^jFollow. up the chain to .oiir^ .final
conclusion. If this mere setting ouTVere the whole story,
how simple it would be! Consider this syllogism, in form
typically Euclidean and deductive:
Major premiss: All professional mathematicians are
muddleheaded.
Minor premiss: The writer of this book is a professional
mathematician.
Conclusion: Therefore the writer of this book is muddle
headed.
Now the conclusion is quite possibly true, and it is cer
tainly the correct conclusion to be drawn from the two pre
misses. But both the major and the minor premisses are
false (the writer of this book is not a professional mathe
matician: heaven forbid! he is only a teacher), and therefore
the conclusion, even if materially true, is logically absurd.
In fact the main source of fallacious reasoning almost always
lies in false premisses. The truth of the conclusion cannot
be more true than the truth of the premisses, and^ a scrutiny
and a rigorous analysis of tfi^ap is tWefore always necessary.
H CRAFTSMANSHIP IN MATHEMATICS
At bottom, all reasoning is much of the same kind, and
it usually turns on the truth or falsehood of the premisses.
Clear thinking is thus indispensable: probabilities have to be
weighed, irrelevant details discarded, the general rules accord
ing to which events occur have to be divined, hypotheses have
to be tested; the general rules once established, the derivation
of particular instances from them is a simple matter.
But in elementary mathematics fer beginners, the_prQiipn
of concrete particular instances comes first in importance.
In the handling of his subject in the classroom, the mathe
matical teacher cannot be too concrete. As the boys advance
from Form to Form, they will gradually begin to understand,
and in the Sixth to realize fully, that the solving of every
mathematical problem consists first of disentangling, then of
setting out and classifying, then of tracing similarities and
finding possible connecting links, then of linking up and
generalizing; in other words, o analysis followed by syn
thesis. Although without generality there is no reasoning,
without concreteness there is neither importance nor signifi
cance. But in schools logical rigour is a thing of exceedingly
slow growth. We shall return to the question of mathematical
reasoning in a later chapter.
More often than not, presentclay writers of standard text
books in mathematics strain after both ultraprecision of
statement and the utmost rigidity of proof. But any attempt
in schools to be perfectly exact all at once, to include in every
statement all the saving clauses and limitations that can be
imagined, inevitably ends in failure. This is where the be
ginner, untrained, just down from the University, so often
blunders. He is inclined to argue that, unless his classroom
logic is as unassailable as that of his University Professor,
his work will be open to serious criticism. The work of even
Sixth Form specialists cannot be placed on an unimpeachable
logical basis. The_ degree of riggur_ r tiiat,caa.,h.jixacted at
apyjrtage must necessarily ^depend on the degree of intellectual
development of^ilie pupil. A school can never become a
place for mathematicaTasceticism,
TEACHERS AND METHODS 15
But, as boys get older, they should be encouraged to read
their own textbooks " up and down, backwards and forwards ".
In their study, let us say, of the calculus, let them first obtain
an insight into general elementary processes, and then pro
ceed at once to simple applications. Ample practice in
differentiation and integration is, of course, necessary, but
the study of geometrical and dynamical applications must not
be unduly delayed. It is these that will excite interest, and
will help greatly to produce an appreciation of fundamental
principles. But again and again go back to a more critical
examination of those principles. The applications will have
taught the boys a great deal of the inner meaning of the pro
cesses, and the more abstract discussions will then be made
much easier by the fact that the learner has acquired a fair
stock of more or less concrete ideas.
The Fostering of Mathematical Interest
The general standard of mathematical attainments in
Sixth Forms is now reasonably satisfactory, and entrants at
the newer universities are beginning work of much the same
grade as entrants at Cambridge. But though Sixth Form
specialists are doing solid work (of a very restricted type, it
must be added), the amount of mathematical work being done
by all the other pupils who have obtained the School Certifi
cate is, as a rule, slight, too slight and much toojacademic for
the fostering,,, of a JifeJong interest in the subject. Let the
younger race of teachers wake up to this important fact, and
help to put things right. We shall refer to this point again.
Books to consult:
1. Didaktik des mathematischen Unterrichts, Alois Hofler.
2. A Study of Mathematical Education, Benchara Branford.
16 CRAFTSMANSHIP IN MATHEMATICS
CHAPTER II
Which Method: This or That?
Old and New. Rational and Ruleofthumb.
An intelligent woman, who is known to have done a fail
amount of mathematics in the days of her youth, recently
received a bill for 8s. 7rf., representing the cost of 7 Ib. 6 oz.
of lamb. She was " sure " that the ounces and farthings had
been included merely for the purpose of cheating her, and
she telephoned to the butcher to know the price of the meat
per Ib. She was quite unable to calculate the amount for her
self (Is. 2d.).
A wellknown Inspector of the nineties dictated this sum
to a class of 11yearolds: " Take one million ten thousand
and one from ten millions one thousand one hundred ". As
might be expected, hardly any children had the sum right.
The Inspector looking grieved, the Teacher gently asked him
if he would himself work the sum on the blackboard. Very
unwisely the Inspector tried to do so, and made a hopeless
mess of it to the delight of the boys.
The first story illustrates ojiejDf the commonest faults of
scBool mathematics: teachers are apt to 'push on into more
advanced work before foundations have been well and truly
laid. The second story shows that a^nonmathematician
should not be allowed to criticize mathematical teacher^. To
the nonspecialist, mathematics is full of pitfalls, and it may
be hoped that the time will come when every teacher of the
subject will be a trained mathematician, even if he has to
teach nothing but elerrientary arithmetic.
Not the least important question for a teacher of elemen
tary arithmetic to consider is the method ...of ._aBPtflacb , tQ . a
new rule. Should that rule be given to the child dogmatically,
given as a rule, to be followed by the working of examples
WHICH METHOD: THIS OR THAT? 17
until it is thoroughly assimilated? or should the rule be
" explained ", approached " intelligently ", and be thoroughly
" understood ", before it is applied to examples? In other
words, is it immoral or is it legitimate to provide a child with
a working tool before the nature of the tool is explained?
To put it another way: suppose that we teach a rule
" intelligently ", and the children get 50 per cent of their
sums right; or suppose that we teach by rule of thumb and the
children get 80 per cent of their sums right. Which plan
should we adopt?
Should we give credit merely for " getting sums right "?
or should we forgive mere slips if the working shows some
grasp of the process?
Again: suppose we find that some of the newer and
popular methods, methods that have superseded those in
common use forty or fifty years ago, are less productive of
speed or accuracy or both, are we, or are we not, justified in
feeling a little suspicious of the newer methods?
Some of these questions have been answered for us by the
psychologists, who in recent years have adopted various
devices for testing the comparative merits of the methods
we use in teaching arithmetic. The old school of psycholo
gists trusted t&cKrnuch to intuition, and their views were
doctrinaire. ^Presentday psychologists, on the other hand,
are devoting themselves to experiment, to_the garnering of
facts, to making careful deductions from those facts. For
instance, some of them have**ar ranged with schools for tens
of thousands of simple sums, of varying types, to be worked
by different methods. From such large numbers of results
legitimate deductions may be drawn, especially when different
psychologists arrive at similar conclusions from different sets
of examples. It is on such evidence as this that different
methods have been compared and some sort of priority
determined. No thoughtful mathematical teacher would now
pronounce dogmatically in favour of. his own method of
doi^ pithing, even if he has used it : all his life. 'pHe would
suEJect it, ancl other methods as well, to prolonged tests
(B291)
18 CRAFTSMANSHIP IN MATHEMATICS
selecting different groups or classes of children all " new "
to the principle to be taught; and he would compare the re
sults in different ways, for instance for intelligence, for
accuracy, and for speed; and he would make sure that the
general conditions of the tests, for instance the time of day
when they were given, were equalized as far as possible. It
is in such matters that psychologists are helping us greatly.
Perhaps the first essential of all is accuracy, especially
Accuracy in all kinds of computation. What would be the
use oT"a bank clerk who^ made mistakes in running up a
column of figures? A tradesman inaccurate in his calculations
might jsoon ^find hunselt abankrupt . Indeed , accuracy jranjjs
as a ^cardinal viytiie^ it Jgja main factor of morality. A boy
who gets a^smnjw rongjshQ.uld~ be made to get it right. Never
accept a wrong answer. This does not mean that credit
shbuldTTot be given for intelligence: anything but that. For
instance, a boy may be given a stiff problem and get it wrong.
But that problem may include half a dozen little independent
sums, each of them to be thought out before it can be actually
worked; five of them may be right and one wrong. In such
a complex operation, a margin of error may be legitimately
allowed for.
If we think of our own personal operations in arithmetic,
those we are engaged in day by day, we must admit that most
of our working is by rule of thumb; the actual rationale of a
process does not enter our heads. We have become almost
mathematical automata. Yet, if called upon to do so, we
couIcCof course, explain the rationale readily enough. But
the average boy, the average boy, however intelligently he
may have been taught, not^only works by rule of thumb but
could not for the life pf jiij&.give~an adequate explanation pf
tEe^ocess. This is admittedly brutal fact. Test any average
class of 30 boys, twelve months after they have been taught
a new rule, and it is highly improbable that more than 8
or 10 will explain the mathematical operation adequately
and intelligently. The experienced teacher never expects it.
Nevertheless, no teacher worth his salt would ever dream
WHICH METHOD: THIS OR THAT? 19
q teaching a new rule j^ithout % approaching _Jt ratio.nally.
He would do his best to justify every step of the process,
illustrating and explaining as simply as possible. Perhaps
4 or 5 of the boys in a class will see the whole thing clearly,
and their eyes may sparkle with satisfaction. A few more,
perhaps 8 or 10, will follow the argument pretty closely,
though if asked to repeat it they will probably bungle pretty
badly. But the rest? No. They want the rule, simply and
crisply put, a rule they can follow, a rule they can trust and
hold fast to. And no teacher need break his heart that the
majority can do no more. No inspector, if he is a mathe
matician, ever expects more; he is too familiar with the
mathematical limitations Jthat nature has imposed on the
average boy; of British origfn^ ~
As a boy goes up the school and his intelligence is de
veloped, the fundamental processes of arithmetic may be
made clearer to him. Any average boy of 13 or 14 may be made
to understand the main facts of our decimal system of nota
tion, whereas at 7 or 8 he may have failed to grasp the real
significance of even a threefigure number. Every teacher
of mathematics should remember that he cannot clear the ground
finally as he goes along\ he has to come back again and again.
Do not worry young children with such terms as abstract
and concrete. Nothing is gained by telling a child to add 8
sheep to 9 sheep instead of 8 to 9. Actual arithmgtical pro
cessejy*e^ll abslrjcjt, j^djyhe^jnotion.^f , jcastingjsvjgry sum
into problem fprm_baL^J?^QQm^ a_illy fetish. Presentday
cEi!3ren are suffering from a smjeUjjf oranges and^apples.
Of course when little children are beginning to count, to add,
to subtract, &c., the use of real things is essential, and in this
matter we may learn much from t he efficient
teacfaer. Some of the very fcest arithmetic J^achJn&J .
school^. It is a pleasure to watch chil
dren who are little more than toddlers getting a real insight
into number and numeration. The worst teaching of arith
metic I have ever seen was in the lower forms of the old
grammar schools of 40 years ago. In those days it was not
20 CRAFTSMANSHIP IN MATHEMATICS
an uncommon thing for the lowest forms to be placed in
charge of an unqualified hack. Those were dark days indeed.
" Practical " mathematicsjincludes manipulative work of
some kind, actiiaf measuring as well as calculating, and the
more of this irTtKT Seconds, Third^ and Fourths the better,
especially if there isTa mathematical laboratory available.
It is concrete mathematics, but do not give it that label. In
fact, put the label into the wastepaper basket. As for the
label abstract, burn it.
Books to consult:
1. The Approach to Teaching, Ward and Roscoe.
2. The New Teaching y Adams.
(These are not books specially directed to mathematics, but to
teaching generally. They are books to be read by every teacher,
for they are full of good things. Mr. Ward was for many
years chief Inspector of Training Colleges; Mr. Roscoe is
Secretary to the Teachers' Registration Council, and was
formerly Lecturer on Education at the University of Birming
ham; Sir John Adams was formerly Professor of Education
in the University of London.)
CHAPTER III
"Suggestions to Teachers"
The Handbook of Suggestions to Teachers, 1928, issued by
the Board of Education, contains useful hints " for the con
sideration of teachers and others concerned in the work of
Public Elementary Schools ". The practical hand is revealed
on every page, and there can be no doubt that the best teach
ing practice known in the country is embodied in it. The
book deals specifically with the regui^ments jof Elementary
Schools as they are likely to be developed during thePnext
few years Infant ^Schools, Junior Schools, and Senior
"SUGGESTIONS TO TEACHERS" 21
Schools, including " Selective " Central Schools; but what
is said about mathematics, especially arithmetic, is equally
applicable to scKoofs oFaU types.
The Board are of opinion that, by the age ofJLJU." a mini
mum course should at least include a thorough groundwork
in notatigin, a knowledge of the %s^J^r_rjjle^^^ied[ to
mpngy, and the ordinary English measures of length, area,
capacity, weight, and jtime; an elementary acquaintance with
vulgar and decimal fractions, together with simple notions
of geom_etrigal form and some skill in practical measure
ments." By that age, " accuracy in simple operations should
in great measure be automatic. It depends first on a ready
Knowledge of tables,, and secondly upon concentration, but
in the case of written work is greatly assisted by neatness of
figuring and clear statement."
(j^The Board contemplate that, in future, the arithmetic of
all Senior Elementary Schools (where the age will extend
from 11 to 14 or 15 or even 16) will be associated with men
Duration, scale ^drawing, geometry, gny>h, and (for boys)
algebra, trigonometry, and practical mechanics.^ The course
of mathematical work mapped out for such schools is par
ticularly suggestive and should be read by mathematical
teachers in all schools.
The Board seem also to contemplate for Senior Elemen
tary Schools some form of mathematical laboratory where
practical work can be done. This "work is to be associated
with the geometry, mensuration, surveying, mechanics, and
manual instruction, and even for the lqyv^r_cla.s.es^.iiseful
hints are given for practical. work in weights and nieasures.
Much of the work which at one time constituted the pre
liminary course of practical physics might be included as
well the use of the vernier and the micrometer screw gauge,
the volumes of irregular solids by displacement, densities
and specific gravities, Utube work, and experimental veri
fications of such principles as those involved in the lever
and pulleys, in the pendulum, and in Hooke's law.
The time has gone by when arithmetic, even in Elemen
22 CRAFTSMANSHIP IN MATHEMATICS
tary Schools, should be looked upon as a selfcontained sub
ject. Although arithmetic is the subject dealing with numerical
relations, it is geometry which deals with space relations, and
tfuTtwo should be taught together. Algebra is just a useful
mathematical instrument, full of
devices for both arithmetic^and geometry. TTrigpnometry
is the ^^ surveyor's subject ,jTTiseful appl^itojp_or^j^bra,afld
geometry together. A graph is a geometrical picture, showing
arithmetical amTalgebraic relations of some sort. The various
subjects fuse tojjether as parts of a single puzzle^and quite
young boys may be given a working insight into them all.
Arithmetic alone is dry bread indeed, far too beggarly a
mathematical fare even for a Junior School.
From the first, keep the mathematical work _jn_close con
tact Ayith^thfe, problems, ^of practical life. Let matters reasoned
about be matters with which the children are either already
familiar or can be made to understand clearly. Po_ not, take
the children for excursions into the clouds^ what is perfectly
cleaFlo you may JBe^veiyJoggy to them.^ Therejs no^ inde
pen3ent " jaculty " oTreagpning, independent of the par
ticular facts and relations reasoned about, stored away ghost
like in the brain, to be called upon when wanted. Hence,
always endeavour to ensure that the things which^ you call
upon a boy to discuss are seen by him as in a polished mirror.
of mathematical terms.
If you take care always to use such terms in tHeir exact sense
they need rarely be defined. Even very small children have
to learn the terms add, subtract, sum, difference, remainder,
whole, part, less, equal, equals, total, and older children m'ust
acquire an exact knowledge of such terms as interest, discount,
gross, net, balance, factor, prime, measure, multiple, and dozens
of others. If you use them consistently, the children will
soon learn to appreciate their exact significance.
Let part of your stockintrade be pricelists of some of
the big London stores, the Post Office Guide, Bradshaw,
Whitaker's Almanack, and the like. Ask yourself what sort of
mathematical knowledge the children are likely to require
" SUGGESTIONS TO TEACHERS" 23
inafter life, and, as far as you can r provide accordingly. But
it_is not merely a^guestion of giving them practicaT tips:
train them to think mathematically. Train them to_car^_Jor
accuracy. Train them to appreciate some of the marvels of
thelnirverse the very great and the very small.
^ not ^ise .^Ijd^sJuoned methods that have stood
the test of time, and do not be too ready to adopt the new
fangled methods of some new prophet. Any new educational
lubricant which is advertised to be a tremendous accelerator
of the classroom machinery generally proves to be nasty
clogging stuff, making life a burden for those who use it.
Book to consult:
Handbook of Suggestions for Teachers, H.M. Stationery Office.
CHAPTER IV
Arithmetic : The First Four Rules
Numeration and Notation. Addition
We have already mentioned that the laborious work of
psychologists has taught us much about the pitfalls experienced
by beginners when learning arithmetic. Few young teachers
realize the number of separate difficulties felt by children in
learning to do ordinary addition sums, even after the addition
table to 9 f 9 is known.
For instance, a child has to learn:
(1) To keep his place in the column;
(2) To keep in mind the result of each addition until
the next number is added to it; and
(3) To add to a number in his mind a new number he
can see;
24 CRAFTSMANSHIP IN MATHEMATICS
(4) To ignore possible empty spaces in columns to
the left;
(5) To ignore noughts in any columns;
(6) To write the figure signifying units rather than
the total number of the column, specially learn
ing to write when the sum of the column is
20, 30, &c.;
(7) To carry.
A teacher should analyse in this way every general arithmetical
operation, and provide an adequate teaching of every separate
difficulty. Unless at least the slower pupils are thus taught,
they may break down in quite unsuspected places. Another
important thing is the grading of difficulties. For instance,
we now know that the average beginner finds the addition sum
21 4
43 easier than 21
35 3
35
and the latter very much easier than 21+ 43 + 35. He seems
to have more confidence in the completed columns ? and the
vertical arrangement appeals more stronglyto his eye^ than
does^T^horizonfal arrangement
Here is a series of first subtraction sums, graded accord
ing to the difficulty experienced by beginners:
888
380
Teach one thing at a time] see to it that this one thing does
not conceal a number of separate difficulties; and let that one
thing be taught thoroughly before the next is taken up.
The bare elements of numeration and notation will have
been taught in the Infant School or Kindergarten School,
and on entry to the Junior School or Department the children
will clearly apprehend the inner nature of a 3figure number,
that, for instance,
357 = 300 + 50 + 7.
ARITHMETIC: FIRST FOUR RULES
25
If that is thoroughly understood, but not otherwise, numeration
and notation should give little further difficulty.
Numbers of more than 6 figures will seldom be required
in the Junior School or Department, and children soon learn
to write down 6figure numbers correctly. Let beginners
have two 3column ruled spaces, thus:
Thousands.
H.
T.
U.
H.
T.
u.
2
4
3
5
9
6
Tell them they have to fill up the spaces under " thousands "
exactly as they fill up the old familiar 3column space on the
right. Dictate " 243 thousands ", and pause; the child
writes 243 under " thousands ". Now go on: " listen to
what comes after thousands; 596 ". The child soon learns
to write down a dictated number of " thousands ", just as
he would write down a dictated number of " books ". With
a class of average children of 10 or 11 years of age, one lesson
ought to be enough to enable them to write down even
9figure numbers accurately, if these are properly dictated,
and if the children are first made to understand that after
" millions " there must always be two complete groups each
of 3 figures, the first of these groups representing thousands.
Millions.
Thousands.
H.
T.
U.
H.
T.
U.
H.
T.
U.
The teacher dictates: " Write down 101 million 10 thousand
and one."
" How many millions?" " 101." " Write 101 under
millions' 9
" How many thousands?" " 10." " Write 10 under
thousands"
26 CRAFFSMANSHIP IN MATHEMATICS
" What comes after thousands?" " 1." " Write 1 in the
righthand 3column space."
1 1
1
" Now fill up with noughts."
Numbers without noughts should come first. Introduce the
noughts gradually. Remember that they provide constant
pitfalls for beginners.
Dispense with the ruled columns as soon as possible,
but let the successive triads always be separated by commas;
14,702,116 (14 millions, 702 thousands, 116).
Subtraction
On entry to the Junior School the child will already have
been taught that the subtraction sum
is a shortened form of
800
100
867
142
725
60
40
700 + 20 + 5
bundles of sticks or bags of counters being used to make the
process clear.
They will also have been taught to decompose the top
line in such a sum as this, leaving the lower line alone:
534
386
Thus:
500 + 30 + 4 __ 400 + 120 + 14
300 j 80 + 6 ~" 300 +80+6
100+ 40+8 = 148
ARITHMETIC: FIRST FOUR RULES 27
I have often seen excellent results in such an instance, the
small children handling their bundles or bags, untying them
and regrouping, in a most businesslike way. They really
did seem to have grasped the essentials of the process.
But can the method be regarded as the most suitable
permanent possession for older children? Consider this sum:
80,003
47,167
The necessary decomposition is a complicated matter for
young children. They have to take 10,000 from the 80,000;
leave 9000 of the 10,000 in the thousands column and carry
1000 on; leave 900 of this 1000 in the hundreds column and
carry 100 on; leave 90 of the 100 in the tens column and
carry 10 on to the units column. And thus we have:
80,000 + 3 70,000 + 9000 + 900490 + 13
40,000 + 7000 + 100 + 60 + 7 40,000 + 7000 + 100 + 60+ 7
30,000*+2000 + 800 + 30+ 6 = 32,836
In practice this is what we see:
7999
** % q q'3
4 7, 1 6 7
"~
I confess that, judged by the number of sums right, the best
results I have ever met with were in a school where this
decomposition of the minuend was taught, although the
teacher responsible was not only not a mathematician but was
entirely ignorant of the principles underlying the plan she
had adopted. She would give the children a sum like this:
70,024
30,578
Before the children began actually to subtract, they had
to examine each vertical column of figures, beginning with
the units column. If the upper figure was smaller than the
28 CRAFTSMANSHIP IN MATHEMATICS
lower, they would borrow 1 from " next door ", prefix it
to the unit figure in the top line, and show the borrowing
by diminishing by 1 the figure they had borrowed from.
Thus they wrote:
l
7 0, *{ 1 4
3 0, 5 7 8
Then they would examine the tens column. If, as before,
they found the upper figure smaller than the lower, they
would borrow from next door again if they could; if not,
they would pass along to the place where borrowing was
possible, in this case 7, change the 7 to 6, and prefix the
borrowed 1 to the 1 in the tens column, always changing
into 9's the O's they had passed over. They always " borrowed
from next door when anybody was at home, putting a 9 on
the door of every house they found empty ". Thus the sum
was made to look like this:
6 9, 9 11
X q ^ 1^4
3 0, 5 7 8
Then the subtracting was begun, and, of course, it was all
plain sailing: 8 from 14, 7 from 11, 5 from 9, from 9,
3 from 6.
Over and over again I tested that class, and not a child
had a sum wrong. But the children had no idea of the " why "
of the process; neither had the teacher. The accuracy was
the result of a clear understanding of an exactly stated simple
rule. The children followed the rule blindly.
But this case does not typify my general experience,
which is that the decomposition method, is not productive
of anything like the accuracy obtained by the alternative
method of equal additions. Although, therefore, I am drivgn
to favour the equal additions method, this method does not
seem quite so susceptible of simple concrete explanation
for very young beginners. Still, such explanation is possible.
First try to make the pupils understand that equal additions
ARITHMETIC: FIRST FOUR RULES 29
to the minuend and subtrahend * will not affect the difference.
The ages of two children provide as good an illustration as
anything:
Jack is 7 and Jill is 10; their difference is 3. In 4 years*
time, Jack will be 11 and Jill will be 14; their difference
will still be 3. And so on. Their difference will always be 3.
A first lesson on brackets will serve to reinforce the idea:
107 =3
(10 f 4)  (7 f 4) = 14  11 = 3
(10 + 6)  (7 + 6) = 16  13 = 3
Then
73 73 + 5 = 78 73 + 10 = 83
21 or, 21 + 526 or, 21 f 10 = 5U
52 52 52
In this way, get the child to grasp the cardinal fact that in
any subtraction sum we may, before subtracting, add any
thing we like to the top line if we add the same thing to the
bottom line; the answer will always be the same. Another
example:
73 70 4 3 70 ^13
48 40 + 8 50+8
Here we have added 10 to the top line, turning 3 into 13,
and we have added 10 to the bottom line, turning 40 into
50. (The double arrow usefully draws attention to the two
additions.) Thus the answer to the altered sum will be the
same as to the original sum. In this way it is easy to give a
clear understanding of the socalled " borrowing " process.
But the small child is not quite so happy when working
by this method in the concrete, as he is with the decomposition
method. When he is given the two extra bundles of 10,
he does not always believe that the sum can be the same.
* Do not use these terms with beginners; wait until the senior school. The /row
line and take line will do, or the top line and bottom line, or the upper line and lower
line. Adopt simple terms of some kind, and adhere to them until the children are
better prepared to adopt a stricter nomenclature.
30 CRAFTSMANSHIP IN MATHEMATICS
However, some kindergarten teachers seem to have little
trouble about it.
In a sum like the following, the teaching jargon * we
should use for beginners would probably be something like
this:
" Whenever wejyjve_l to thetop line, we musLalwapgive
1 to the bottom line as well, but next door."
9023
3765
" 5 from 3 we cannot; give 1 to the top line and so turn
3 into 13; 5 from 13 is 8. Now give 1 to the bottom line,
next door; 1 and 6 is 7."
" 7 from 2 we cannot; give 1 to the top line, and so turn
2 into 12; 7 from 12 is 5. Now give 1 to the bottom line,
next door; 1 and 7 is 8."
" 8 from we cannot; give 1 to the top line and so turn
into 10; 8 from 10 is 2. Now give 1 to the bottom line,
next door; 1 and 3 is 4.
" 4 from 9 is 5."
The words borrow and pay back tend to mislead the
slower boys, since we borrow from one line and pay back
to another. To them this seems unfair, especially when we
say we borrow 10 and pay back only 1.
Personally I prefer to give 1 to the top line and never
talk about paying back, but compensate by giving 1 to the
bottom line. But the 2 parts of each double transaction must
be worked in association at once; this satisfies the children's
sense of justice.
In each of the first several lessons, ask what the giving
of 1 really signifies. " When we turned 3 into 13, the 1 given
was really 10; did we give the bottom line the same number?
* The term jargon is rather suggestive of slang, but, of course, what I really mean
is the simple homely language which we mathematical teachers all invent for teaching
small boys, language which rather tends to offend the ear of the English purist. But
that does not matter. The important thing is to express ourselves in words which
convey an exact meaning to the children's minds,
ARITHMETIC: FIRST FOUR RULES 31
Yes, because the 6 which by adding 1 we turned into 7
is in the tens column.
" When we turned the 2 into 12, the 2 was really 20,
and the 1 we gave to it was really 100; did we give the bottom
line the same number? Yes, because the 7 which by adding
1 we turned into 8 is in the hundreds column."
And so on. A very nnsui era r>le proportion of the children
will not at this stageunderstand the process at all. But do
not worry about that. Come back to it in a Ygsy^Jimg*
It will be weeks, even months, before the slower child
will have had enough practice to do subtraction quickly and
accurately, and it is best to adhere all the time to precisely
the same form of teaching " jargon ".
There remains the question, shall we teach the children
(i) actually to subtract, or (ii), to add (complementary ad^
ditionj, or (iii), first to subtract from 10 and then add the
difference to the figure in the top linel For instance,
13
Shall we say 5 from 13 is 8 i
or, shall we say 5 and 8 is 13? '
or, shall we say 5 from 10 is 5 and 3 is 8?
The last must be ruled out of court; it is productive of great
inaccuracy amongst beginners, though later on it is useful in
money subtraction. The jsecond^Elan is popular, but it has
been proved to be less productive of accuracy than thejiffi;
and it is Something ot' a sham, for thenumber to be added
must be obtained by subtraction.* Hence the first method,
honest subtraction, is strongly advocated, and that demands
ample practice in both the subtraction and the addition
tables. Thus the child learns:
both
5 and 1 is 6
5 and 2 is 7
5 and 3 is 8
and
6 from 6 is 1
6 from 7 is 2
5 from 8 is 3
&c.
* Cf . algebraic subtraction.
32 CRAFTSMANSHIP IN MATHEMATICS
"5 from 8?" is as effective a form of question as (and is
much more elegant than) " 5 and what makes 8?".
The Tables
The child must learn the addition table to 9 + 9 perfectly.
He must be able to say at once that, e.g., 9 and 8 is 17, He
must also be able to say at once that 9 froni^rLiSA and
that 8jTom_jL7 is 9. In fact, the addition and subtraction
tables should be learnt in close association. Very young
children when learning to count, to add and subtract, will,
of course, be shown how to find out that 8 + 311 and
that 11 3 = 8, but the time must soon come when they
can give those results pat, without calculation or thinking
of any kind; and this means a great deal of sheer dingdong
work from which lower forms and classes can never escape.
Never mind the charge of unintelligence; be assured that
the people who make such a silly charge have never had to
face the music themselves. Table accuracy is the one key
tp_ accurate arithmetic.
^EaclTsubtraction table is, of course, as already indicated,
the mere complement of an addition table. For instance,
the 4 times addition table begins 4 and 1 is 5 and ends 4
and 9 is 13; the corresponding subtraction table begins
4 from 5 is 1 and ends 4 from 13 is 9. Carry the addition
tables to 9 + 9 and the subtraction tables to 18 9.
How many repetitions are necessary to ensure permanent
knowledge? All experienced teachers know that this varies
enormously. It may be that only 10 repetitions are required,
but it may be 500, according to the individual. Test, test,
test, day by day. Do not waste the time of a whole class
because further drill is necessary with a few.
Helpful blackboard tests may be given in a variety of
forms, e.g.,
9 + 6 = * 17 4 = #
6 + 8 = x 13 + 8 = x &c.
In examples of this kind we have the germ of equations,
ARITHMETIC: FIRST FOUR RULES 33
as we had with the examples in brackets. Explain that x
is a symbol for the number to be found. Call on a member
of the class and point to the first x, then call on another member
and point to the second x. But do not call on members in
order. Keep every child in expectation. Call on Smith the
shirker half a dozen times a minute. If the answers are not
given at once, without any calculations, the tables are not
known, and more drill is necessary.
Draw a circle of small numbers on the board and have
them added together, as they are pointed to. The answers
must be instantaneous or the tables are not known.
Make the children count forwards and backwards, by
1's, then by 2's, then by 3's, &c.
1, 4, 7, 10, 13, 16, &c.
100, 96, 92, 88, 84, 80, &c.
This sort of practice helps the tables greatly.
But do not expect that, because a boy knows 7 + 6 is 13,
he will therefore know that 27 + 6 = 33. Such extended
examples require special practice, and the practice must
be continued day by day until the boy knows at once that a
7 added to a 6 always produces a 3. Similarly with sub
traction; a boy must be able to say at once that a 7 taken from
a 6 always produces a 9.
Write on the blackboard, say, a 7. " Let us add 6's."
Smith? 13; Brown? 19; Jones? 25; dodging about the class.
The response must be instant. Similarly with subtraction.
Let your schemes for testing the tables be as varied as
possible. Do not be satisfied as long as there is a single
mistake. Do not forget that dull boys may require 10 times,
perhaps 50 times, the practice that quick boys require. There
must be no counting on fingers, no strokes, no calculations of
ainy kindT
So with the multiplication and division tables. Beginners
are taught, of course, that multiplication is just a shortened
form Q a succession of additions, and dm 8 * 011 a _ shortened
fofm^of a succession of subtractions. That fact grasped,
fB~29n " " 4
34 CRAFTSMANSHIP IN MATHEMATICS
then come the tables, multiplication to 9 X 9 and division
to 81 f 9.
Do not be intelligently silly and teach a boy " to find
out for himself " the value of 9 X 8 by making him set out
9 rows of 8 sticks each and then count to discover 72. Make
him learn that 9 X 8 ==72. _ When he begins multiplication
and division, a few very easy concrete examples will be given
him, to make the fundamental ideas clear. Then make him
learn his tables, learn his tables.
As with the addition and subtraction tables, write the
multiplication and division tables side by side. The sign
for " equals " may well be substituted for " is ".
1X7=7 7's into 7 = 1
2 X 7 = 14 7>s into 14 = 2
3 X 7 = 21 7's into 21 = 3
9 X 7 = 63 7's into 63 = 9
Mental work:
Seven threes?
Three sevens?
Sevens into twentyone?
Threes into twentyone?
Let the 3, the 7, and the 21
hang together in all 4
ways.
And so on.
Ask for the factors of such numbers as 42, 77, 28, &c.
Blackboard Work:
Write down a number consisting of 15 or 18 figures,
and ask the class to give the products of successive pairs of
figures, as rapidly as possible: e.g.
371498652498.
Answers:
21, 7, 4, 36, 72, &c.,
Again:
7 X 3 = x\ 3 X 7 = x\ 3 X x = 21; 7 X x = 21; x X 3 = 21;
x X 7 = 21.
21 21
= x\ = x,
37
ARITHMETIC: FIRST FOUR RULES 35
And so on. Point to an #, and call on a particular pupil for
the answer.
Mental work in preparation for multiplication and division
sums:
(3 x 7) + 1 = x. (3 x 7) + 2 = x (8 X 7) + 5  x.
3's into 22 = x. 3's into 23 = x. 8's into 61 = x.
Ample practice in this type of example is necessary. The
examples are of course one step beyond the simple tables;
there are two operations, one in multiplication or division,
one in addition or subtraction. Hence instantaneous response
is hardly to be expected from slower children. But it is
surprising how quickly the answers come from children
who know their tables, who know that 8 X 7 = 56 and that
56 + 5 = 61, though it is well to remember that a mental
effort is required to keep in mind the first answer while it
is being further increased or diminished.
The 11 times table is hardly worth learning. The 12 times
table may^ be ^ostponed^ until money sums are taken up.
The 15 times is easy to learn and is useful for angle division.
So is the 20 times table. Mental work on simple multiples
is easy to provide, e.g. 18 X 9 = twice 9x9; (16 X 7) =
twice (8x7).
But when actually teaching the tables, it is a safe rule
not to complicate matters by giving tips for exceptional cases.
Do not, for example, tell a beginner that, when he is adding
a column of figures, he should look ahead to see if two of
them added together make 10. If he has to find the sum of
4, 8, 3, 7, 5, teach him to say, 4, 12, 15, 22, Jj7^not to look
ahead and to discover that 3 + 7 = 107~and then to say
4, 12, 22, 27. Such a plan with beginners makes for in
accuracy. Good honest straightforward table work must
come first. Short cuts may come later, when they may be
more readily assimilated.
36 CRAFTSMANSHIP IN MATHEMATICS
Multiplication
It is easy to make any average child who is well grounded
in numeration and notation understand that 4 times 273
means the sum of four 273's, i.e.
273
273
273
273
and that therefore the answer is
(4 times 200) + (4 times 70) + (4 times 3);
and he sees readily enough that the teacher's shortened
form
273 200 70 3
4 444
1092 = 800+280+12
But the slower child will not understand, though he will
learn the ordinary rule of multiplication fairly readily.
In teaching multiplication, the advisable succession of
steps seems to be:
(a) Easy numbers by 2, 3, and 4; no carrying; no zeros
in multiplicand.
(b) Easy numbers by 2, 3, and 4; no carrying; zeros in
multiplicand.
(c) Easy numbers by 2, 3 ... 9, with carrying; no zeros
in multiplicand.
(d) Easy numbers by 2, 3 ... 9, with carrying; zeros in
multiplicand.
(e) The same with larger multiplicands.
(/) Multiplication by 10.
(g) Multiplication by 2figure numbers not ending in a
zero.
ARITHMETIC: FIRST FOUR RULES 37
(h) Multiplication by 2figure numbers ending in a zero,
(i) Multiplication by 3figure numbers, zeros varied.
Be especially careful to show clearly the effect of multiplying
by 10, viz. the shifting of every figure in the multiplicand one
place to the left in the notational scheme, i.e. each figure is
made to occupy the nextdoor position of greater importance.
Then show the effect of multiplying by 100, by 1000, by
20, 200, 6000, &c. Bear in mind that the work has particular
value, inasmuch as ultimately it will lead on to decimals.
From the outset, use the term multiplier and the term
product, but let the difficult term multiplicand wait until the
senior school stage. The term top line will do for juniors.
When we come to ordinary 2figure and 3figure multi
pliers, which of the following processes is preferable, the
first or the second?
34261 34261
43 43
102783 1370440
137044 102783
1473223 1473223
The first is the oldfashioned method; the second is newer and
at present is popular. The second is often advocated because
(1) it leads on more naturally to the rational multiplication
of decimals, (2) it is preferable to multiply by the more
important figure first, if only because the first partial product
is a rough approximation to the whole product.
The first reason does not appeal to me at all, for I am
very doubtful about the allied method of multiplication of
decimals. The second reason is undoubtedly a good one.
Numerous tests of the comparative merits of the two
methods have shown that the old method leads to a much
greater accuracy than fE new, and to me that seems greatly
to outweiglTthe advantage of the new method. Slower boys
seem to have much more confidence in a method where they
have to begin with both units figures, as they do in addition
38 CRAFTSMANSHIP IN MATHEMATICS
and subtraction. In any case I deny that the newer method
is " more intelligent " than the old7
Division
Begin by instructing the children to write down in standard
division form such little division sums as they know from
their tables. Teach them the terms dividend, divisor, and
quotient: we can hardly do without them.
2(6 3\9 37 59
3 3 2, and 1 over 1, and 4 over.
Now teach them the use of the term remainder, and to write
the letter R for it.
59 46
T, R 4 I, R 2
Now 2figure dividends, within the tables they know.
436 847 979
^9 , R 7 J, R 7
Now 2figure dividends beyond the tables they know.
" 4's into 93? the tables do not tell us. Then let us take our
4 times table further:
10 x 4 = 40
11 X 4 = 44
23 x 4 = 92
24 X 4 = 96
" Evidently 4's into 93 are 23, and 1 R. Hence
493
23, R 1
ARITHMETIC: FIRST FOUR RULES 39
" But we need not have written out that long table; we may
work in this way:
" 4's into 9? 2 and 1 over; write down the 2.
" By the side of the 1 over, write down the 3, to make 13.
" 4's into 13? 3, and 1 over. Write down the 3.
" The last 1 over is our Remainder.
" But what does this mean? When we said 4's into 9 we
really meant 4's into 90, and when we wrote down the 2,
the 2 really meant 20. Here is a better way of showing it
all, and we will write the figures of the answers above the
dividend, instead of below it.
4)93
8_ = 80 = 20 times 4
13
12 = 3 times 4
" First we took from the 93, 20 times 4, and had 13 left.
" Then we took from the 13, 3 times 4, and had 1 left.
" Altogether we took from the 93, 23 times 4, and had
1 left."
A little work of this kind will suffice to justify the process
to the brighter children; a few will grasp it fully. The dullards
will not understand it all; they want the clearcut rule, and
explanations merely worry them.
Now consider an ordinary long division sum; say, divide
45329 by 87. Let the children write out the 87 times table,
to 9 X 87.
1 X 87 = 87
2 x 87 = 174
3 x 87 = 261
4 x 87 = 348
5 X 87 = 435
6 X 87 = 522
7 X 87 = 609
8 X 87 = 696
9 X 87 = 783
(In making a table like this note that 3 times = 2 times
40 CRAFTSMANSHIP IN MATHEMATICS
+ 1 time, 5 times = 3 times + 2 times, &c., and so save
the labour of multiplying; only multiplication by 2 is necessary;
all the rest is easy addition.)
87)45329
" 87's into 4? won't go: 4 is not big enough; put a dot
over it.
" 87's into 45? won't go: 45 is not big enough; put a
dot over the 5.
" 87's into 453? will go, because 453 is bigger than 87.
How many times?"
Look at the table, and take the biggest number (435) that
can be subtracted from 453. The 435 is 5 times 87. Place
the 5 over the 3 in the dividend, write the 435 under the 453,
and subtract; the difference is 18.
Bring down the 2 from the dividend, placing it to the right
of the 18, making 182. Look at the table again, and take the
biggest number (174) that can be taken from the 182, &c.
"521
87)45329
435! :
~T82;
174:
"^9
87
" Thus we know that 87 is contained 521 times in 45329,
and that there is 2 (the Remainder) to spare.
" What is the biggest R we could have? Could it be 87?
Why not?"
Teach the children the usual verification check: multiply
the divisor by the quotient, add R to the product, and so
obtain the original dividend.
(Do not forget, when introducing formulae later, to utilize
the D, d, Q, and R. D = dQ or dQ + R.)
ARITHMETIC: FIRST FOUR RULES 41
Now pour a little gentle scorn upon making out a special
multiplication table for every division sum: " We must
give up such baby tricks ". But that leads us to what some
beginners in division find very difficult how to tell the
number of times a big divisor will go into one of the numbers
derived from the dividend:
69)342
" Instead of saying * 69's into 342 ', we cut off the last figure
of the 69 and of the 342 and say 6's into 34 instead. This
seems to be 5, but the 5 may be too big, because of the carry
figure; we find it is too big, so we try 4 instead. "
Warn the children that if, after subtracting at any step,
they have a difference greater than the divisor, the figure
they have just put into the quotient is too small. Rub this
well into the dullards.
Warn them, too, that, above every figure in the dividend,
they must place either a dot or a figure for the quotient.
"341, R = 267.
329)112466
Similarly in short division except that the dots and figures
go below:
7)13259
1894, R = 1.
A dot is not a very suitable mark, owing to confusion with a
decimal point; it is, however, in common use. / If no mark is
used, figures get misplacedand errors made. The marks
may be~ dropped later. ~~~
The advantage of the method of placing the quotient
over jnstead of totheright of the dividend, jsjliat children
write down noughts when these
are reguired^. '
Eet division by factors stand over until the senior school.
The calculation of the remainder is puzzling to beginners.
Divide 34725 by 168. Suitable factors of 168 : 4 X 6 x 7.
42 CRAFTSMANSHIP IN MATHEMATICS
434725 units.
8681 fours, R = 1 unit.
u OUOA iuurs, r\ = JL unit.
7 ' 1446 twentyfours, R = 5 fours.
' 206 one hundred and sixtyeights, R = 4 twentyfours.
Total Remainder = (24 x 4) + (4 X 5) + 1
= 117.
Quotient = 206, R =* 117.
Avoid the Italian method, except perhaps with A Sets.
With average children the method is productive of great
inaccuracy.
In fact, avoid all short cuts until main rules are thoroughly
mastered. For instance if a boy has to multiply by 357,
do not teach him to multiply by 7, and then multiply this
first partial product by 50 to obtain his second partial product;
it is simply asking for trouble.
Of course, practised mathematicians do these things, but
we have to think of beginners. Teach a straightforward
method, and stick to it. Hints as to " neat dodges " and
about " short cuts " are for the few, not for all.
CHAPTER V
Arithmetic : Money
Money Tables
No part of arithmetic is more important than the various
manipulative processes of money. It is with us every day
of our lives, and accuracy is indispensable. The ordinary
money tables must be known, and thus more dingdong work
is necessary. This is mainly a question of a knowledge of
the 12 times table. Five minutes' brisk mental work twice
ARITHMETIC: MONEY 43
a day will pay, sometimes with and sometimes without the
blackboard, and sometimes on paper.
Associate with the 12 times table:
1 x 12 = 12
2 X 12 = 24
3 X 12 = 36
I2d. = Is.
24<*. = 2*.
36 d. = 3s. , &c. &c.
Day by day drill:
SQd. = ? 83<f. = ? S4d. = ? 89<f. = ? &c.
Pence in 7s. ? in 7s. 3d. ? in 9s. 9J. ? &c.
and so every day until accurate answers up to 144*/. are
instantaneous. If the boys are familiar with two definite land
marks in each " decade ", 20 and 24, 30 and 36, 40 and 48,
&c., the" necessary additions for the other numbers of each
decade are simple.
Associate the farthings table with the 4 times table, and
the shillings table with the 20 times table, which is easily
learnt.
Let every mental arithmetic lesson at this stage include
simple addition and subtraction of money, especially the
addition of short columns of pence.
Elementary facts concerning the coinage should be
associated with the money tables, and in this connexion do
not forget guineas (which often figure in subscriptions and
in professional fees) and Bank of England notes.
At a later stage the boys should be taught such common
place facts about the coinage as every intelligent person
ought to know, e.g. the nature of " standard " gold and
silver, degrees of " fineness ", the nature of the present
legalized alloy for " silver " coinage, the nqyket prices of
pure gold and silver, the nature of bronze; alloys; tokens;
the Mint.
Reduction
Reduction is not likely to give serious trouble, if the
tables are known. The commonest mistake is to multiply
44 CRAFTSMANSHIP IN MATHEMATICS
instead of divide, or vice versa. Teach the boy to ask himself
every time whether the answer is to be smaller or larger\
if smaller, to divide; if larger, to multiply. But " guineas to
pounds ", and the like, is a type of sum that baffles the slow
boy and requires special handling.
Subtraction
There is something to be said for avoiding, at first
writing farthings in the usual fractional form and for giving
them a separate column:
* d. f.
47 14 6 1
21 19 4 3
25 15 1 ~2
let the children use the fractional forms a little later, when
they may be made a useful introduction to fractions. The
alternative is to omit farthings altogether in the early stages.
Multiplication
How is this to be done? Whatever method is adopted,
a percentage of inaccurate answers seems almost to be in
evitable. We set out a sum by each of the 4 methods commonly
used. Multiply 7, 15$. 10W. by 562.
I. s. d.
7 15 10J
10
77 18 9 = 7 15 10J X 10
10
779 7 6 = 7 15 10J X 100
5
3896 17 6 = 7 15 10J x 500
15 11 9 = 7 15 10J x 2
467 12 6 = 7 15 10J X 60
4380 1 9 = 7 16 10J X 562
ARITHMETIC: MONEY
II.
662
16
20 84305.
421,105. Od.
12[6620</.
20 468^.
2562/z.
23, 85. 4d.
1228U.
235. 5d.
45
7
X
662 =
3934
16
X
662 =
421
10
10
X
562 =
23
8
4
t
X
562 =
i
3
5
7
16
10*
X
562 =
4380
1
9
III.
562 at 7, 155.
7
lojd.
3934
=
7
X
562
*
ofl
281
=
10
X
562
t
of 10/
140
10
=
5
X
562
of 5/
17
11
3 =r
7J
X
562
i
tv
of 5/
7
6 =
3
X
562
4380
1
9 =
7
15
iot
X
562
IV.
7, 165. 10H x 562
= 779375 x 562
= 43800875
= 4380, 15. 9d.
My own experience, and this corresponds to the results of
many inquiries, is that the second method produces the best
results; then the third (" practice ") method, provided that
pupils have been well drilled in aliquot parts (though some
always seem to find division more difficult than multiplication);
then the first method. The last is a good method for older
pupils who have learnt to decimalise money readily, but
not for younger pupils or for slower older pupils.
The ordinary method (the first method) is curiously
productive of errors; in the course of a long experience I
have never known a whole class, without exception, get a
sum right by this method, even after they had had several
months' practice. The second method generally leads to
untidy and unsystematic marginal work. This marginal
46 CRAFTSMANSHIP IN MATHEMATICS
work should be made an integral part of the working of the
sum, and should not be looked upon as scrap work.
Division
Dr. Nunn's suggestion that the process of working may
be set out in the following way may well be followed. All
pounds are kept in one vertical column, shillings in another,
and so on. It is very neat and compact. Allow plenty of
space across the paper. Example: Divide 3541, 14s.
by 47.
s. d. f.
75 7 1 1
47)3541
329
14 9 2
>320 ( >60 ,88
251
334
69
90
235
329
47
47
16 X 20
5 X 12
22 X 4
43 R
Answer: 75, 7s. \^d. and 43 farthings over.
Other methods have been devised, but this old method is
probably best and most readily learnt.
To ensure a full understanding of the nature of the
" remainder " a sum like the above should be followed up
by two others:
1. Take 43 farthings from the dividend; then divide
again.
2. Add 4 farthings to the dividend; then divide again.
Even slower boys can usually explain the (to them) rather
surprising new answers.
WEIGHTS AND MEASURES 47
CHAPTER VI
Weights and Measures
Units and Standards
Consider what weights and measures are used, to what
extent, and how, in practical life. Teach these, and teach
them well, and let all the rest go. A coal merchant concerns
himself with tons, cwt., qr., never with Ib. and oz.; a grocer
with cwt., qr., Ib., oz., never with drams and rarely with tons;
a farmer with acres, quarteracres, and perches; a builder
with yards, feet, and inches; a surveyor with chains and links;
and so generally. The completer tables of weights and
measures are generally given in the textbooks as a matter
of convenience, but, in practical life, the whole table is
seldom wanted by any one person. A teacher who gives boys
reduction, multiplication, or division sums, say, from tons
to drams (or even to ounces), or from square miles to square
inches, is simply proclaiming aloud his incompetence: perhaps
he is the slave of some stupidly written textbook; certainly
he is lacking in judgment. The main thing is to make the
boys thoroughly familiar with the few weights and measures
that are commonly used, and to give them a fair amount of
practice in the simpler transformations of comparatively small
quantities; and to let all the rest go.
Teach clear notions of units and standards. Show how
unintelligent we British people have always been in our
choice of units. We have, for instance, determined our inch
by placing three grains of barley in line; we have selected
our foot, because 12 of the inches roughly represent the
length of a man's foot; we have determined our smallest
weight (the grain), by adopting the weight of a dried grain
of wheat. That such things vary enormously did not trouble
our forefathers at all. Tell the boys that at one time the
French people had similar unsatisfactory weights and measures
48 CRAFTSMANSHIP IN MATHEMATICS
but that now they have changed to a system much more
rational.
Let the various tables be learnt and learnt perfectly.
Weight
1. Avoirdupois (not used for the precious metals). Let
the table to be learnt include the oz., Ib., qr., cwt., ton.
Note that the standard weight is the pound, which consists
of 7000 grains. (A dried grain of wheat, though roughly a
grain in weight, is not, of course, a standard. A grain is
1/7000 part of a pound.)
Teach the stone as a separate item: normally 14 lb., but for
dead meat, 8 Ib.
Give easy sums for practice in:
(1) tons, cwt. y qr. (coal and heavy goods).
(2) cwt., qr., Ib. (wholesale grocery).
(3) cwt., st., Ib. (wholesale meat purchases).
(4) Ib., oz. (retail grocery and meat).
Note that an oz. of water or any other fluid is an avoirdupois
ounce, like the ounce of any common solid, and contains
7000/16 or 437 grains.
In making up arithmetical examples, utilize as far as
possible the quantities (sacks, bags, chests, &c.) representing
the unit purchases of tradesmen and others, though the
problem given will often depend on the locality. For instance,
problems on crans and lasts of herrings, or on trusses of hay
or straw, would be quite inappropriate in big inland towns.
The teacher should, for problem purposes, make a note of
points like the following: weight of a chest of tea, f cwt.;
sack of coal or of potatoes, 1 cwt.; bag of flour, 1J cwt.;
bag of rice, 1 cwt.; truss of straw, 36 Ib.; truss of new hay,
60 Ib., of old hay 56 Ib.; a brick, 7 Ib.; 1000 bricks, 3 tons;
100 Ib. of wheat produces 70 Ib. of flour which produces
91 Ib. of bread; and so on. Everyday quantities of this kind
WEIGHTS AND MEASURES 49
give a reality to problems in arithmetic that even the non
mathematical boy appreciates.
2. Troy, used by jewellers. Let the table to be learnt
include the grain, dwt., oz., Ib. It is important to remember
that the Troy ounce is heavier than the common (avoirdupois)
ounce, since it contains 480 grains, as against 437^. But
the Troy pound is lighter than the common pound, since it
contains only 12 ounces and therefore 5760 grains, as against
7000 in the avoirdupois Ib.
3. Apothecaries'. The old weights have gone out of use.
Drugs are generally used in very small quantities, and the
basic weight is the grain (the grain is a constant weight for
all purposes). A quantity of drugs weighing more than a
few grains is expressed as a fraction of an ounce avoirdupois.
N.B. Ignore the avoirdupois dram (^ oz.) and the
druggists' old scruple and drachm weights. The dram and
drachm were not the same.
Length
Let the main table to be learnt include the in., ft., yd.,
pole, fur ., mile, and let the link, chain, and fur. be included in
a separate table. Remind the boys that the chain is the
length of a cricket pitch.
Give easy sums for practice in:
(1) yd., ft., in. (builders, &c.).
(2) poles, yd., ft. (farmers, &c.).
(3) miles, chains, links (surveyors).
(4) miles, yd. (road distances, &c.).
Measures that may be drawn from practical life for
problem use are almost innumerable. The sizes of battens,
deals, and planks will be learnt in the manual room; notes
of the sizes of other materials used by builders slates,
glass, doorframes, &c., &c. may be made from time to time;
size of an ordinary brick, 8f " X 4" X 2" (note the \" all
(E291) 5
60 CRAFTSMANSHIP IN MATHEMATICS
round for jointing), square tile, 9f" X 9f " X 1" or 6" X 6"
X 1"; machineprinted wallpaper, 11 \ yd. X 21"; hand
printed, 12 yd. X 21"; French, 9 yd. X 18"; sheets of
paper, foolscap, 17" X 13" (see Whilaker for other sizes);
bound books, foolscap 8vo, 6f" X 4" (see Whitaker)\
skein of yarn = 120 yd., hank = 7 skeins; railway gauge
4' 8" (12' of roadway for single track, 23' for double); equator,
24,902 miles; polar diameter, 7926 miles; fathom, 6'; knot,
6080' (40 knots == 46 miles). These are only a tithe of the
everyday measurements that may be used for making up
problems. Such problems are far more valuable than the
hackneyed reduction sums given in the older textbooks.
N.B. The ell, league, and such foreign lengths as the
verst, may be ignored. The cubit is worth mentioning.
Area
Let the table to be learnt include the sq. in., ft.,, yd. pole,
the rood, the acre, sq. mile. It is useful to remember that an
acre = 10 sq. chains, or a piece of ground 220 yd. X 22 yd.,
or a piece just about 70 yd. square.
Give easy sums for practice in:
(1) sq. miles, acres (areas of counties, &c.).
(2) ac., ro., sq. poles (farmers, &c.).
(3) sq. yd., sq.ft., sq. in. (builders, &c.).
Familiar areas for problem purposes: Lawn tennis
court, 78' X 36' or 78' X 27'; Association football ground,
120 yd. X 80 yd.; Rugby, 110 yd. X 75 yd.; croquet lawn,
105' X 84'; Badminton court, 44' X 20'; &c.
Volume
Table: c. in., ft., yd.
Let sums for practice be of the simplest, e.g. the number
of cubic yards of earth excavated from a trench; the number of
WEIGHTS AND MEASURES 51
cubic feet of brickwork in a wall; the cubic capacity of a
room or of a building; the volume of the Earth in. cubic
miles.
Capacity
1. Liquids. Table: gill, pt., qt., gall. Casks have a
variety of names: barrel of ale 36 gall.; hogshead of ale
54 gall., of wine = 63 gall., &c. A wine bottle = gall.;
Winchester quart = \ gall.
2. Dry Goods (corn, &c.). Table: peck, bushel, quarter.
The quarterpeck is called a " quartern "; the halfpeck is
the equivalent of a gallon. The gallon is a kind of link between
the liquid and dry measures.
There is now a tendency to substitute weight for measure.
Problems on capacity reduction are hardly worth doing,
except small problems that may be done mentally. But
problems involving transformations between capacity and
weight are common, and ample practice is necessary. N.B.
1 gall, of water weighs 10 Ib. " A pint of pure water weighs
a pound and a quarter."
Liquid medicine measure (mainly solutions in water).
Table: 60 minims = 1 fluid drachm.
8 fluid drachms = 1 fluid ounce.
20 fluid ounces = 1 pint.
The fluid ounce is the same as the common (avoirdupois)
ounce, and therefore weighs 437^ grains. But it contains
480 minims, and therefore a minim weighs rather less than
a grain. The minim may be thought of as a " drop ", though
of course drops vary greatly in size.
Doctors' prescriptions may be discussed, rather than
sums worked. If a solid drug is prescribed, the amount is
expressed in grains or in fractions of an ounce; if liquid, then
minims, drachms, or ounces.
52 CRAFTSMANSHIP IN MATHEMATICS
Time
The second, minute, hour, day, week, give little trouble.
The variable month requires careful explanation. Teach
the doggerel " Thirty days hath September ", &c., or furnish
some alternative mnemonic. Explain " leap " year and its
determination.
Few problems of reduction are necessary, and these should
be easy. A few on the calendar are advisable, and a few
dealing with speeds.
Useful Memoranda
Other useful memoranda for problemmaking. (The quan
tities are approximations only and should be memorized.
They are useful when closer approximations have to be
evaluated):
1 cubic foot of water 6J gall. 62 J Ib.
1 cubic inch of water = 252 grains.
A common cistern 4' x 3' X 2* ' = 30 c. ft. 187 gall. = ton.
1 ton of water = 36 c. ft. = 224 gall.
1 gallon of water = 277 c. in. = 10 Ib.
1 ton of coal occupies about 40 c. ft. (hence 25 tons need a
space 10' X 10' X 10').
Wall Charts
A few permanent charts are useful on a wall of the class
room where weights and measures are taught: an outline
plan of (1) the town or village showing the overall dimen
sions, length, breadth, and area; (2) the schoolsite and
buildings; (3) the school itself; (4) the actual classroom.
(5) Diagram to scale to show that 5 yd. X 5 yd. =
30 J sq. yd. (often used for a first lesson in fractions).
(6) A chart giving the weights of a few familiar objects in
and about the school, and the capacities of a few others. See
that these charts are used and known.
WEIGHTS AND MEASURES 53
The Metric System
Some knowledge of this system is necessary, if only
because of the work in the physical laboratory. The be
ginner may be shown a metre measure side by side with a
yard measure, and simply be told that it is rather longer,
and had its origin in France. As the boy goes up the school
he will learn that its length is about 3937 in., and is the
measured fraction of a quadrant of the earth's surface. A
little later still, he will be told how the French measured
the actual length of an arc of one of their meridians, and
how they determined the latitude of each place at the end of
the arc. This easy astronomical problem is usually worked
out in a Fifth Form geography lesson.
The cubic decimetre and the litre, the cubic centimetre
and the gram, are, as derivations of the initial metre, always
a source of interest to boys.
The boys should memorize the few usual approximate
equivalents between the British and metric systems, e.g.
1 metre = 3937 in.; 1 kilogram = 22 lb.; 1 litre = 176
pints; 1 gram = 1543 grains; 1 are = $ acre. With these
they can quickly estimate quantities in terms of metric units.
For instance, a Winchester quart will hold 4/176 litres
= 227 litres = 2270 c. c.; 1 hectare = 2 acres; and so on.
But do not forget to enter a defence in favour of our own
system of weights and measures, if the metric system is
advocated on purely scientific grounds. Sixth Form boys
are always interested in this. In the first place, the metre
was not measured accurately; in the second place, it is a
local and not a universal unit; it depends upon the length
of a particular meridian in a particular country. The meridian
the French measured was an ellipse, not a circle, and not
a true ellipse at that. Had they utilized the polar axis (a
fixed length) instead of a meridian (a variable length), their
unit would have been more scientific, for it would have been
universal, and it could have been measured more accurately.
54 CRAFTSMANSHIP IN MATHEMATICS
The length of the polar axis is very nearly 500,500,000 in.,
so that the inch already bears a simpler relation to the polar
axis than the metre does to its own meridian quadrant. If we
adopted a new inch, viz. 1/500,000,000 of the polar axis,
it would make but a very slight change in our linear measure
ments, and then, curiously enough, a cubic foot of water
would weigh almost exactly 1000 oz. (instead of 997). Our
present ounce weight would have to be increased by only
Y part of a grain! Moreover, the new cubic foot would
contain exactly 100 halfpints. Such a new system would
be incomparably more scientific than the metric system.
Thus the opponents to the adoption of the metric system
have sound arguments to support their views. The metre
has on its side the virtue of being the basic unit of a con
venient and simple system; but scientifically it is a poor
thing.
There is no need for the boys to learn the metric tables.
But they should learn the three Latin prefixes deci, centi, milli,
and know that these represent fractions; and the three Greek
prefixes deca, hecto, kilo representing multiples. These learnt
thoroughly, the tables as such are unnecessary. But with
three or four exceptions the multiples and submultiples are
hardly ever wanted.
The best exercises on the metric system are those based
on laboratory operations.
CHAPTER VII
Factors and Multiples
The term " factor " and " multiple " should be used
when the tables are being taught, though without formal
definition. " 3 X 7 = 21; we call 3 and 7 factors of 21."
Give me a factor of 6? 3; another? 2; a factor of 30? 2;
FACTORS AND MULTIPLES 55
another? 3; another? 5, A multitude of people means
many people, and a multiple of a number means a bigger
number containing it many times, though " many " may not
be greater than 2. Now think of your 5 times table. Give
me a multiple of 5? 15; another? 30; another? 35. After a
little of this work, the terms factor and multiple will become
part of the boys' familiar vocabulary. " Common " factor
and multiple will come later. One idea at a time.
Tests of Divisibility
Prime Factors. Tests of divisibility for 2, 3, 5, and 10^
may readily be given in the Junior School or Department;
those for 4, 8, 9, 11, 12, 25, 125, a year or two later. At first,
give the rules dogmatically.
" A number is divisible by 2 if it is an even number.
3 if the sum of its digits is
divisible by 3.
5 if it ends in a 5
10 if it ends in a 0."
Justification, not " proofs ", of such rules is commonly
given in Form IV. The reasoning, which is quite simple,
depends on the principle that a common factor of two
numbers is a factor of their sum or their difference. Never
mind the general proof; at this stage merely justify the
principle by considering a few particular instances, and
these readily emerge from the multiplication table; for
instance:
5 fours = 20
7 fours = 28
12 fours = 48
9 fives = 45
7 fives = 35
2 fives = K)
We know that 5 fours added to 7 fours make 12 fours, i.e.
4 is a factor of 20 and of 28, and is also a factor of 20 + 28.
Again, we know that 7 fives from 9 fives is 2 fives, i.e.
66 CRAFTSMANSHIP IN MATHEMATICS
5 is a common factor of 35 and 45, and is also a factor of
45  35.
Divisibility by 2. Consider any even number, say 754;
754 = 750 + 4. Since 2 is a factor of 10 and therefore of the
multiple 750, and is also a factor of 4, it is, by our rule, a
factor of 750 + 4 or 754.
Divisibility by 5. Consider any number ending in 5,
say 295; 295 = 290 + 5. Since 5 is a factor of 10 and there
fore of the multiple 290, and is also a factor of 5, it is, by
our rule, a factor of 290 + 5 or 295.
Divisibility by 3. Consider any number, say 741.
741 = 700 } 40+1
= (100 x 7) + (10 X 4) + 1
= (99 X 7) + 7 + (9 X 4) + 4 + 1
= (99 X 7) + (9 X 4) { 7 f 4 + 1
= (99 X 7) + (9 X 4) + 12.
Now 3 is a factor of 9 and therefore of all multiples of 9;
it is also a factor of 12. Since 3 is a factor of 99 X 7 and of
9x4 and of 12, it is a factor of their sum, i.e. of 741.
Hence, &c.
A formal proof of the principle used should be associated
with the algebra later.
The justification of the rule for 4 and 25, 8 and 125,
and 9 is equally readily understood, but that for 11 is a
little more difficult.
Primes and Composite Numbers
Quite young boys quickly see the distinction between a
prime and a composite number and are always interested
in the sieve of Eratosthenes.
Third Form boys should be made to memorize the squares
of all numbers up to 20; 13 2 = 169; 17 2 = 289; &c. (The
squares of 13, 17, and 19 must really be learnt; 14 2 , 16 2 ,
and 18 2 can be mentally calculated in a second or two, if
forgotten.) Then give a little mental practice in extracting
FACTORS AND MULTIPLES 57
square roots: of 81? of 256? of 361? (Mention that the root
sign (vO we use is merely a badly written form of the initial
letter R.)
Make the class write down, in order, the successive pairs
of factors of, say, 36:
2 X 18
3 x 12
4x9
6X6
I 9 X 4
112 x 3
\18 X 2
Then point out that the second column is the first column
reversed, and that the 3 lower horizontal lines are the 3
upper horizontal lines reversed. Hence when we have to
write dowu the factors of 36, we need not proceed beyond
the fourth line, viz. 6x6, for then we already have all the
factors; and the 6, the last trial number, is \/36. The boys
can now appreciate the common rule: When resolving a
number into factors, it is unnecessary to carry our trials beyond
its square root, unless the number is not a perfect square,
and then it is advisable to consider the next square number
beyond it.
For instance, write down all the factors of 120; 120 is
not a square number, but the next square number is 121,
the square root of which is 11. Hence we need not proceed
with our trial numbers beyond 11, but as 11 does not happen
to be a factor, we do not proceed beyond 10. Thus by trial
we find that 2, 3, 4, 5, 6, 8, and 10 are factors; and, dividing
120 by each of these, we obtain other factors which pair off
with them. The 14 factors of 120 are
2 3 4 5 6 8 10
60 40 30 24 20 15 12
The next step is to teach factor resolution by trials of
prime numbers only. This causes no additional difficulty,
but the boys should recognize at once all the prime numbers
68 CRAFTSMANSHIP IN MATHEMATICS
up to, say, 41 (1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,
37, 41).
" Write down the factors of 391. The next square number
beyond is 400. Hence we need not carry our trials beyond
V/400 or 20. By trial we find that the lowest factor is 17.
By division we obtain 23, which we recognize as another
prime number. Thus 17 and 23 are the only factors. "
I have known a C Set of a Fourth Form become expert in
factorizing 3figure numbers, after one lesson. It is work
that most boys like.
" Express 360 as a product of factors which are all prime.
Divide by the lowest prime number, 2 if possible; and
again and again if necessary; then by the next prime, 3,
if possible; and again and again, if necessary. Then by 5,
if possible; then 7; then 11; and so on.
360 = 2 x 180
= 2 X 2 X 90
= 2x2x2x45
= 2x2x2x3x15
= 2x2x2x3x3x5.
Obviously we now have all the factors, though our trial
division did not proceed beyond 3.
" A neater way of writing down the prime factors of 350 is
2 8 x 3 a x 5.
The little 3 at the top righthand corner of the 2 shows
the number of twos and is called an index.
" Express 18900 as the product of factors which are all
prime.
18900 = 2x2x3x3x3x5x5x7
= 2 2 x 3 3 X 5 2 x 7.
We read, 2 squared into 3 cubed into 5 squared into 7.
An index serves as a useful means of shortening our written
work."
At this stage two or three minutes' brisk mental work
occasionally will help to impress upon the pupils' minds the
FACTORS AND MULTIPLES 59
values of the lower powers of the smaller numbers: 2 3 , 2 4 ,
2 5 , 3 2 , 3 3 , 3 4 , 4 2 , 4 3 , 4 4 , 5 2 , 5 3 , 5 4 , &c.
Common Factors
" Give me a common factor of 36 and 48: 2; another?
3; another? 4; another? 6; another? 12. Which is the
greatest of these common factors? 12. We call 12 the Greatest
Common Factor of 36 and 48. If we write down the prime
factors of the different numbers, we can almost see the G.C.F.
at once.
36 = 2x2x3x3.
48 = 2x2x2x2x3.
Evidently 2 is a common factor of both numbers, and
another 2, and a 3. Hence, the G.C.F. == 2 x 2 x 3 == 12,
i.e. 12 is the greatest number that will divide exactly into 36
and 48."
" It is neater to write down the factors in the index form.
What is the G.C.F. of 540, 1350, 2520?
540 = 2 2 x 3 3 x 5 1 .
1350 = 2 1 X 3 3 X 5 2 .
2520 = 2 3 X 3 a X 5 1 X 7
We see that 2, 3, and 5 are factors common to all three
numbers; from the indices we see that one 2, two 3's, and
one 5 are common. Hence the G.C.F. is 2 1 X 3 2 X 5 1 = 90.
Note that we write down each prime factor that is common
and attach to it the smallest index from its own group. "
However clear the teaching, I find that there is usually
a small number of slow boys who are puzzled by the index
grouping. Hence in lower Sets the extended nonindexed
groups of factors are preferable. Always sacrifice a neat
method if it leads to puzzlement and inaccuracy.
60 CRAFTSMANSHIP IN MATHEMATICS
Common Multiples
' Give me a multiple of 5: 25; another? 35; another?
55. Give me a multiple of 3: 21; another? 15; another?
60. Give me a common multiple of 3 and 5: 60; another?
15; another? 30. Which is the least of all the common
multiples of 3 and 5? 15; i.e. 15 is the smallest number into
which 3 and 5 will divide exactly. We call it the Least
Common Multiple^
" Find the L.C.M. of 18, 48, and 60."
Write down the numbers as products of their factors,
expressed in primes.
18 = 2 X 3 X 3.
48 2x2x2x2x3.
60 =2x2x3x5.
The L.C.M. , being a multiple of the three given numbers,
must contain all the factors of the numbers, but it must
not contain more, or it will not be the least common multiple.
" 1. The L.C.M. has to contain all the factors of 18;
write them down as part of the answer:
L.C.M. = 2 x 3 x 3 x
" 2. In order that the L.C.M. may contain all the factors
of 48, it must include four 2's and one 3. We have already
written down one 2; hence we must write down three more.
As we have already written down two 3's, another is not
necessary. Hence,
L.C.M. =2x2x2x2x3x3x
" 3. In order that the L.C.M. may contain all the factors
of 60, it must include two 2's, one 3, and one 5. We already
have two 2's, and a 3, but no 5. Hence we must include
a 5. ^ < ^ ^ ^ ^ ^/
L.C.M. = 2x2x2x2x3x3x5
= 720.
FACTORS AND MULTIPLES 61
" It is neater to write down the factors in the index form.
What is the L.C.M. of 54, 72, 240?"
64 = 2 X 3 3 .
72 = 2 3 X 3 2 .
240 = 2 4 X 3 1 X 6 l .
We mayjvritc down the L.C.M. at once^, by writing downjwerx
one of the prime^ factors and attacEmg to_gach the greatest
index ot its group:
L.C.M. = 2 4 x 3 s x 5
= 2160
It is well to provide pupils with some little mnemonic,
to enable them to keep in mind that:
the smallest index concerns th^ greatesJ^C^.
and the greatest index concerns the fegg^CJVI^
There is much to be said for using the terms greatest
and smallest (or least) in arithmetic, and the terms highest
and lowest in algebra. The former terms are obviously
correctly applicable to magnitudes. Beginners do not
find it easy to appreciate the exact significance of highest
and lowest. The term measure is best avoided. The dis
tinction between it and factor is a little subtle for boys.
The oldfashioned division methods of G.C.F. and L.C.M.
are cumbrous and unnecessary, and slower boys never un
derstand the processes, the formal " proofs " of which are
quite difficult enough for Fifth Forms. Numerical illustrations
of the principle that a common factor of two numbers is a
factor of their difference may be utilized to justify the ordinary
G.C.F. procedure, if the procedure itself is considered
necessary.
62 CRAFTSMANSHIP . IN , MATHEMATICS
CHAPTER VIII
Signs, Symbols, Brackets. First
Notions of Equations
Terminology and Symbolism
Mathematical terms should always be used with precision;
then formal definitions in all early work will be unnecessary.
Sum, difference, product, quotient are terms which should be
quite familiar even to Juniors; they are the A, B, C of the
whole subject. So should the signs + , , X, and .
Multiplicand and multiplier, dividend and divisor should also
become current coin at an early stage, though there is difference
of opinion about the first term in this group. I am not quite
sure about subtrahend and minuend, even in the senior school;
they are commonly confused. If we bear in mind the English
significance of the Latin nd (gerundive), the nd terms
can be explained in a group.
Multiplicand, is a number that has to be multiplied.
Dividend is a number that has to be divided.
Subtrahend is a number that has to be subtracted.
And of course minuend is a number that has to be minused
or reduced, but boys will confuse minuend and subtrahend.
If the terms are used let subtrahend come first, and minuend
a good deal later.
As for the division sign f, hammer in the fact that the
dots stand for numbers, that when we write, e.g.,  we mean
4 divided by 5, and that we might write out our division table,
either
51 = 5
10 f 2 = 5
15 ~ 3 = 5
Y =
since both mean exactly the same thing.
Algebraic letter symbols may be introduced at a very
SYMBOLS, BRACKETS, EQUATIONS 63
early stage. (Do not look upon arithmgtjc^giijd algebra^as
cUstantjcouin&, but as twin brothers, children to be brought
up together.} Begin with the simple consideration of lengths
and areas. Establish by a few numerical examples that the
area of a rectangle may be determined by multiplying length
by breadth. Select rectangles whose sides are exact inch
multiples, ignoring all fractions until later. Then introduce
the notion of a " formula " a convenient shorthand means
of keeping an important general arithmetical result in our
mind. " We have found that however many inches long,
and however many inches broad, a rectangle is, the area in
square inches is equal to the product of the inches length
and the inches breadth. It is easy to remember this by taking
the first letter of the word length (/), of the word breadth
(6), and of area (A), and writing the result so:
/ x b = A.
But we generally save time by writing Ib = A, omitting the
multiplication sign. Always remember that when in algebra
two letters are written side by side, a multiplication sign
is supposed to be between them. Instead of the letters /,
b y and A, any other letters might be used."
Rub in well the principle taught, giving a few simple
evaluations.
Now consider a square area, / X /, or m X m\ II or mm.
(Distinguish between inches square and square inches.)
Follow this up with cases of rectangular solids, and
establish such formulae as V = I X b X h = lbh\ then the
cube, V = aaa.
" When we were working factors, we adopted a plan for
shortening our work. Instead of writing 4 X 4 X 4, we wrote
4 3 , the little 3 at the top righthand corner (which we called
an index) showing the number of 4's to be multiplied together.
So in algebra.
aa may be written a 2 ,
aaa may be written 3 .
Then what does a 5 mean? a 3 i 2 ?" And so on.
64 CRAFTSMANSHIP IN MATHEMATICS
Avoid all difficult examples at this stage. The main thing
is to teach the new principle. Keep the main issue clear.
Let hard examples wait.
Brackets
" A pair of brackets is a sort of little box containing
something so important that it has to receive special attention.
The brackets generally contain a little sum all by itself. If
I write
9 + (7 + 3)
or 9 f (7  3),
I mean that the answer to the little sum inside the brackets
has to be added to the 9. If I write
 (6 + 2)
or 9  (5  2),
I mean that the answer to the little sum inside the brackets
has to be subtracted from the 9.
" Now I will work out the four sums:
9 + (7 + 3) = 9 + 10 = 19.
9 + (7  3) = 9 f 4 = 13.
9  (5 + 2) = 9  7 = 2.
9 _ (5 _ 2) = 9  3  6.
Are the brackets really of any use? Let us write the same
sums down again, leaving the brackets out, and see if we
get the same answer:
9 4 7 + 3 = 19
9 + 7  3 = 13
9  5 f 2 = 6
9  5  2 = 2.
The first two answers are the same, the last two are not. But
look at the last two again. It looks as if they had been changed
over. Thus
9 (5 + 2) is the same as 9 5 2,
and 9 (5  2) is the same as 9  5 + 2."
SYMBOLS, BRACKETS, EQUATIONS 65
With a few easy examples like this, we are in a position to
justify the rule that a f sign before a bracket does not affect
the + and signs within, but that a sign before a bracket
has the effect or converting + and signs within to and
f~~ respectively^ Thus you are able to give the rule and to
justifyit! That is enough at present. Give enough easy
examples to ensure that the rule is known and can be applied
with certainty. " Proof " should play no part at this early
stage. The algebraic minus sign comes later.
Now show the effect of a multiplier.
4(6 f 3) = 4 X 9 == 36.
" We might have multiplied the two numbers separately
in this way,
4(6 + 3)  24 + 12 = 36,*
and when the brackets contain both letters and numbers we
must do it in that way:
5(N f 3) = 5N + 16
for we cannot add N to 3."
First Notions of Equations
Simple equated quantities. For convenience at this early
stage we may call the following an equation:
7 + 5 = 21  9.
Establish the fundamental fact about an equation that we
may add to, subtract from, multiply, or divide each side of
an equation by any number we like, provided that we use
the same number for both sides. Give several examples,
to illustrate each of the four operations. To enable the class to
* A repetition of the sign of equality in the same line should never be allowed
in school practice; it is almost always ambiguous. We do it sometimes in this book
merely to save space.
( E 291 ) 6
66 CRAFTSMANSHIP IN MATHEMATICS
see the operations more clearly, put the original quantities
in brackets. Thus:
(7 + 5) + 4 = (21  9) + 4
(7 + 5)  4 = (21  9)  4
4(7 + 5) = 4(21  9)
7 + 5 219
Do not talk about " proofs "; you are merely verifying
particular instances, to enable the boys to see that your rules
are not arbitrary but are based on reason. A little practice
in such easy examples as the following may usefully follow.
5 times a certain number is 65. What is the number?
" We have to find a certain unknown number. Let us call
it N. The sum tells us that
5N == 65.
Divide each side of the equation by 5; then, N = 13, the
number we require."
The class is not quite ready for such an example as the
following, but they can follow out their teacher's reasoning,
and their appetite is whetted.
Divide 32 into two parts, so that 5 times the smaller is
3 times the greater.
" The two parts added together must make 32, so that
one part taken from the 32 must give the other.
Let S stand for the smaller number.
Then 32 S must represent the greater.
The sum tells us that
6 times the smaller == 3 times the greater.
So we may write
6S = 3(32  S).
SYMBOLS, BRACKETS, EQUATIONS 67
Removing the brackets, by multiplying by 3,
5S = 96  3S.
We cannot see the value of S from this, because we have S's
on both sides of the equation. But, adding 3S to each side,
we have:
6S + 3S == 96  3S + 3S
8S  96
S = 12, the smaller number
and 32 S = 20, the greater number.
Now let us verify the results;" &c.
CHAPTER IX
Vulgar Fractions
First Notions of Fractions
Vulgar or Decimal Fractions first? The first notions of
vulgar fractions will be given in the preparatory Forms,
where the significance of at least halves and quarters will be
understood and the manner of writing them down known.
In the lower Forms of the senior school, it is probably wise
first to give a few lessons on the nature and manipulation of
vulgar fractions, then to proceed with decimals, and to
return to the more difficult considerations of vulgar fractions
later.
The first thing is to get clearly into the child's mind
that a fraction is a piece of a thing, a piece " broken off " a
thing. Take one of several similar things (sticks, apples),
and break off or cut off a " fraction " of it. Cut one of the
things into 2 equal parts, and introduce the term halves;
into 3, and introduce the term thirds; into 4, and the term
68 CRAFTSMANSHIP IN MATHEMATICS
quarters; and see that the terms halves, thirds, quarters,
fifths, &c., are made thoroughly familiar. " I have cut this
apple into 8 parts: give me 1 eighth; give me 5 eighths. I
add 2 of the eighths and 5 of the eighths together: how many
eighths have I?"
We have a special way of writing down fractions. We
draw a line; under it we write the name of the parts we cut
the apple into, over it we write a figure to show the number
of the parts we take: thus
3 4
fifths ' sevenths
Parts of the same name may be added together. Just as we
say
2 apples + 3 apples = 5 apples,
so we may say,
2 sevenths } 3 sevenths = 5 sevenths,
and we write,
23 5
sevenths sevenths sevenths
Let the child see clearly that the fraction shows
number of parts
name of parts '
and, a little later on, introduce the terms numerator and
denominator:
numerator
denominator'
where num = number and nom = name. If the children
learn Latin, give the Latin words.
Then come to fractions of collections of things:  of the
class of children, J a basket of apples, ^ of a Ib. of cherries.
The way is now paved to fractions of mere numbers: of 32;
^ of 27; and so on. But at this stage avoid the terms abstract
and concrete.
VULGAR FRACTIONS 69
For illustrating fractional processes, every teacher will
utilize concrete examples of some kind drawn from everyday
lifeT "As the number 60 contains numerous^ easy factors,
fractions of a crown (60rf.) and of an hour (60 minutes)
make good examples for mental work.
Mental work may profitably be undertaken as soon as
the nature of a fraction is fully grasped.
" Number of pence in  of I/? in ? in ? in ? in
yr 2 ? in ^ ?" Let the children thus discover that f = ^  = j^,
that different fractions may therefore have the same value.
Thus we come to the notion of " cancelling " and its con
verse.
Again:
" How many minutes in 3 of an hour? 20.
i ? 1 r )
4 *''
1 ? 10
5
Thus (5 + J + i) ot an hour 47 minutes.
How may we express 47 minutes as the fraction of an
.'. * + i + 2 = U
hour?
Can you see that this is true? No.
" Well, we have seen that } of an hour = 20 minutes,
and since 1 minute = ( . ] ( j of an hour, 20 minutes = ;? of
an hour; &c. Hence we may write ^ + \ + ^ in this way:
eS + IS" + ^
and wow it is easy to see why the answer is ^.
" Thus if we want to add fractions together, w^jaust
firstsee that they are fractions^ol the same name, i.e. that
they have the same denominator.
" But how are we to change fractions of different denom
inators to fractions of the same denominator ?" And sojvve
cometo L.CJVI.s, &c.
On" the" whole, however, I prefer to illustrate fractional
70
CRAFTSMANSHIP IN MATHEMATICS
processes by means of diagrams, rectangles rather than lines.
A rectangle is conveniently divided up into smaller rectangles
by lines drawn in two directions, and thus the fraction of
a fraction is easily exhibited. We append a few diagrams.
(A squared blackboard or squared paper is always advisable.)
DIAGRAMMATIC ILLUSTRATIONS
Cancelling. ^ = f = f .
X*
Addition. J + i +  = A + A + A =
M
*
*
Fig. 2
Subtraction. f / 2  = f$.
A E
Fig.i
These figures illustrating subtraction will puzzle the slower
children, but squared paper and scissors will soon help to
make things clear.
Multiplication
Multiplication by a fraction is always a little puzzling
at first. A child naturally expects a multiplication sum to
VULGAR FRACTIONS
71
produce an answer bigger than the multiplicand. It is best
to begin with mixed numbers.
The child knows that 2/ X 3 is 6/; by 4, is 8/; and by
3, is 7/. Hence multiplying by the seems to him some
how to have been a real multiplication, inasmuch as the 6/
has been increased to 7/. The multiplication may be con
sidered, as usual, as an addition, viz. of 3 florins and a half
florin. Multiplying a florin by  is to take the half " of " a
florin. Give other examples to show clearly the meaning of
the word " of " when we speak of multiplying by a fraction.
Example: Multiply 2 sq. in. by 3.
Fig. 4
The figure shows 2 sq. in. one above the other, then
another 2 sq. in., then another, then \ part of 2 sq. in. The
last piece shows multiplication by a real fraction, viz. 2 in.X^,
i.e. the strip is \ of 2 sq. in. Thus the whole figure is 6
sq. in. Hence 2 x 3J = 6.
Another example: Draw a figure to show 3J sq. in.\ then
show this multiplied by 2J.
S'4 5? In
9.
,
....
SyJe*.'.
.?*..
r '
Fig. 5
72 CRAFTSMANSHIP IN MATHEMATICS
" How from these figures may we obtain an answer to the
sum 3J x 2? Let us first think of money, say shillings,
instead of inches.
3Js. = 3s. 3d. = 39rf.
Twice 39<*. = 78^.; onethird of 39J. = I3d.
/. 2J times 39J. = (78 + 13) J. = 9 Id. = 7*. 7d. = 7^5.
Apparently, then, 3^ X 2 = 7 1 1 2 . Thus, if we divide the
last figure up into twelfths, we ought to see whether this
result is true. Fig. 6 shows that it is true; the squares need
not be true squares; oblongs will show the fractions just as
well.
Fig. 6
Number of twelfths in the figure:
In the 6 big squares or oblongs, 12 each = 72
In the 2 strips at the sides, 3 each = 6
In the 3 strips at the bottom, 4 each = 12
In the small strip at the corner = 1
Total 91
Thus in the figure we have 91 twelfths = f = 7, as
expected."
Such a figure is satisfactory for illustrating the multiplica
tion of mixed numbers, but for multiplication purposes
mixed numbers should rarely be turned into improper
fractions. A different example is therefore advisable.
Draw a figure to show f of . Lead up to the necessary
figure by showing, first, f; then, by dividing the thirds into
VULGAR FRACTIONS
73
Fig. 7
7 parts, show y of \\ lastly, show  of f. Obviously, now,
4 v 2 __ _8
7 * :i "~ 21"'
" We have found out that:
(i), 2 x 3J, or { X 7,  2 ^;
and (ii), 3] x 2, or \ ;J  X J, = *?;
and (iii), i X }= / t .
Now look: in every case the numerators multiplied together
give the numerator in the answer, and the denominators
multiplied together give the denominator in the answer."
Now the teacher is in a position to enunciate the rule.
He has done nothing to prove the rule, but he has justified
it, so far as it can be justified with beginners.
Division
Just as a child naturally expects a multiplication always
to produce an increase, so he expects a division always to
produce a decrease.
" When you have divided a number by another, the
dividend is always made smaller. Do you agree?" Yes.
" Always smaller?" Yes. " Quite certain?" Yes.
" Let us divide 36 pence amongst some boys.
I give them 12 d. each: how many boys? 3.
od.
2d.
Id.
12.
18.
ob.
Do you mean to say I cannot divide the 36 pence
74
CRAFTSMANSHIP IN MATHEMATICS
amongst more than 36 boys?" Yes, if you give them less
than Id.
" Then give them %d. each. How many?" 72.
" Then 36 divided by J is 72. Thus, although I have
divided 36 I have a quotient bigger than 36. So you were
wrong!" And so on.
A suitable scheme of diagrammatic division is easily
devised, but it is best approached by the division of whole
numbers.
When we divide 24 by 4, the quotient is 6, and the
4 sixes may be arranged in 4 lines thus:
Fig. 8
The shaded section denotes the quotient (6 units); it is a
row of units in line with 1 of the units of the divisor, 4. Any
other row would have done equally well, for any other row
would have been in line with 1 of the units of the divisor, 4.
So generally; a rectangle representing the dividend may
always be divided up in such a way that each horizontal row
of units represents the quotient; there are as many horizontal
rows as there are units in the divisor. Opposite any unit
of the divisor (we select the first) is a horizontal row of units
representing the quotient.
We show 12 divided by 2, by 3, and by 4:
Fig. 9
VULGAR FRACTIONS
75
Let the first fractional problem be to divide 4 by 1.
We may ask how many times I^s. is contained in 4s., i.e.
how many times 16d. is contained in 54d. We may show
this division in the ordinary way, f or  8 ^, which is equal
to 3f . Thus the answer to the sum is 3$. How are we to show
this in a diagram?
We will first divide 3, represented by a rectangle of 3 sq. in.,
by 4, by 3, by 2, by 2, by 1J, by 1, by f.
3r 4= }
3  2J = 
(Divide AB into 5, and take
2 parts for the unit.)
3 7 1J = J 3^1 = 3
(Divide AB into 3, and take
2 parts for the unit.)
Fig. 10
Examine the six diagrams, and note how the quotient (the
shaded part) increases as the divisor diminishes. If then
we diminish the divisor further, the quotient (the shaded
part) must be still bigger. As before, the shaded quotient
must occupy a space opposite one complete unit of the
divisor. But in this case AB is not long enough to show a
76 CRAFTSMANSHIP IN MATHEMATICS
complete unit, only  of one. So we must extend it to make
3. 4 16
7f 4
Fig. 1 1
We may now return to our original example: Divide
by 1J.
Fig. 12
One more example: Divide f fry f .
A
Fig. 13
The first figure shows 1 sq. in.
The second figure shows f sq. in., the part to be
divided by f .
The third figure shows AB extended to one complete
unit, since AB itself represents only f of one.
The last figure shows the result of the division, viz.
. 3 8
~ 4 ^9
VULGAR FRACTIONS
77
We may now collect up our results:
5 fl
2 = 5
4 __ 1 5_
4 ii 7
;3 ~~ "s"
3 _
4 1
An average class will soon discover that by inverting the
divisor the quotient is then obtained by treating the sum
as a multiplication sum, e.g.
But let the teacher be under no delusion. Only a very small
minority will, at the time, appreciate the purpose of the
diagrams. Over and over again I have seen a majority com
pletely baffled, even with very skilful teaching. No matter.
Come back to the demonstration again, a year or two later.
You have justified your rule as far as you can. Now state
it in clear terms and for the present be satisfied that the
boys are able to get their sums right. There is probably
nothing more difficult in the whole range of arithmetic
than the division of fractions, i.e. for boys to understand
the process when it is first taught.
The following kind of argument is sometimes useful:
To divide a fraction, say 4, by 5 is the same thing as
taking of 4. But to take I " of " 4 is the same thing
as multiplying 4 by *; i.e.
4J + 5 = 4J x }.
Now, if we divide 4 by ^, we divide it by a number 7 times
as small as when we divided by 5; therefore our answer
must be 7 times as large as before; i.e.
Since 4J ~ 5 = 4 x J
i.e to divide by  is the same thing as multiplying by J.
Hence, once more, the rule of inverting the divisor.
But the argument is quite beyond the average beginner,
as every experienced teacher knows.
78 CRAFTSMANSHIP IN MATHEMATICS
Cancelling. When a boy is told he may " cancel
thus:
2 5
he is likely to ask, why? He will already have learnt:
(1) That, e.g., 7 X 9  9 X 7;
(2) Reduction of fractions to their lowest terms;
(3) Multiplication of fractions.
Thus he will understand that
If x = 44 x 25
46 22 ~~ 45 X 22 '
and that this may be written:
44 X 25 44 25
_ _ __ QJi _ ^ _
22 x 45 22 45'
He now readily sees that he is justified in reducing each
of these to its lowest terms, and that the final result is the
same as when he cancelled terms in different fractions.
CHAPTER X
* t
Decimal Fractions
A Natural Extension of Ordinary Notation
If care is taken to teach the inner nature of decimal
notation thoroughly, decimals need present little difficulty.
" A decimal fraction is merely a particutyr kind of vulgar
fraction, viz. one with a denominator 10 or power of 10,
DECIMAL FRACTIONS 79
e g Tb T<io> Hoo> Toto o But we do not generally write
them this way; we write them as follows:
7, 73, 8192, 0003
" Let us consider a number consisting entirely of ones
(any other figure would do equally well).
The number 11 = 10 + 1
The number 111 = 100 + 10 + 1
The number 1111  1000 + 100 +10 + 1.
The 1 of least importance in each number is the 1 on the
extreme right; each 1 to the left is 10 times as important
as, and is 10 times the value of, its righthand neighbour.
The 1 on the extreme right represents just one unit.
" But we often break up a unit into parts, e.g. a sovereign,
or a bag of nuts. These parts are fractions, and we might
carry our ones to the right, to represent these fractions,
devising some means of separating the fractions from the
whole numbers: a straight line would do.
Thus, 11111 may stand for 1000 + 100 + 10 + 1 + ^
and, 11111 may stand for 10 + 1+ ^ + t ^ +
Any sort of separating mark will do. Generally we use the
smallest possible mark, a dot, written halfway up the height
of the figures.
" Thus 276347  200 + 70 + 6 + ^ + jfo + T ^.
If we add the 3 fractions together, we get nrcnr Thus
276347 ~ 276 v
" Note the two ways of writing down the same thing:
~
_ 4 _L_ 5 i i 1 _ 4501
i o ~r TOCT ~r Twer ~r ~
__ _
i o ~r TOCT ~r Twer ~r unnnr ~ ioooo
" Perhaps there are no tenths, and we begin with
hundredths:
042 =
We could not write this 42 because 42 Y
80
CRAFTSMANSHIP IN MATHEMATICS
Thus we may have one or more noughts between the decimal
point and the " significant " figures.
" But noughts on the extreme right hand of a decimal
have no meaning:
034 = A + T?
1 ' 1 i
as before/'
Give ample practice in conversion and reconversion, until
the change can be written down mechanically:
37063 == 37^^; ^g^ r 0421; &c., &c.
'conversion of either decimal form into the other is essential.
4 JJ4
iodo 1000
T(JCT(J +
o _ 34
1(7(50(1(1 ~ fooo
This is tfie key to a
Let the rules for conversion be stated in the simplest
possible words.
A few exercises of the following nature are useful:
" If x 10, find the value of these expressions, writing
down the answers in both decimal forms (do not cancel as
you would vulgar fractions):
3 +  +  ? 2 = 34 , 4 + Jff = 3 t Vo, or 347.
x x 2
6r> + 4* + 9 j ~ + ~ = 6049 1 ggg , or 50490306.
Multiplication and Division by 10, by 100, &c.
" If we multiply 347 by 10 we obtain 3470, the 7 becoming
70, the 40 becoming 400, and the 300, 3000.
Th.
H.
T.
u.
3
4
7
3
4
7
Every figure is moved one place to the left, and its value is
increased 10 times.
DECIMAL FRACTIONS
81
" If we divide 2180 by 10 we obtain 218, the 2000 be
coming 200, the 100 becoming 10, and the 80, 8.
Th.
H.
T.
u.
2
1
8
2
1
8
Every figure is moved one place to the right, and its value
is diminished 10 times.
" So it is with decimal fractions, or decimals as we often
call them."
Multiply 3*164 by 10.
3164 x 10 = (3 + t V + T n +
= 30+11 ;' + 1(
= 31 f Vo 3164;
X 10
i.e. the decimal point has been moved one place to the right,
and every figure occupies a place 10 times as important as
before.
So 5623 X 100 = 5623;
005623 x 10000 = 5623.
+
Divide 3164 by 10.
3164 X = (3+ ,',, + ,
= > 3 <> + i w +
'M (14 _
1 O (7
i.e. the decimal point has been moved one place to the left,
and every figure occupies a place reduced in importance
10 times.
Give a number of varied examples in both multiplication
and division, and help the pupils to deduce the rules.
Give plenty of mental work of the following kind:
tens X tens = hundreds
hundreds X tens = thousands
hundreds X hundreds == ten thousands
tenths X tenths = hundredths
hundred ths X tenths = thousandths.
(B 291 ) 7
82 CRAFTSMANSHIP IN MATHEMATICS
Continue this kind of work until instant reponse is obtained
as to the significance of moving the decimal point so many
places to the right or so many to the left. Let the notation
be mastered; then the rest will give little trouble.
Addition and Subtraction
Do not forget the common cause of inaccuracy, blanks
in the fractional columns, especially if the numbers are
arranged horizontally, e.g. 7612 + 31 f 20151.
Multiplication
72314 X 32
_ 79 31 4 vy 32
~~
__ 7 'J 3 1 4 v 3 2
 TT700 X lOtf
_ 2 3 1 4 t H
 10 (TO GO
= 2314048.
The whole process resolves itself into (1) conversion, (2)
multiplication of whole numbers, (3) reconversion.
Note that " conversion " does not mean conversion to
vulgar fractions, but to the alternative form of decimal
fractions, with denominators consisting of powers of 10.
The multiplication of the denominator is really nothing
more than the mere addition of noughts, and it is obvious
from this multiplication that the number of decimal places
in the product is equal to the sum of the numbers of decimal
places in the multiplicand and multiplier, and that from the
very nature of the case this must always be so.
Hence the simple rules:
1. Ignore the decimal point and perform the multiplication
as if the multiplicand and multiplier were whole numbers.
2. Add together the decimal place in the multiplicand and
multiplier; this gives the number in the product. Fix the
point by counting back that number of places from the
right.
DECIMAL FRACTIONS 83
72314
32
144628
21 6942
2314048" No. of dec. places = (3 f 2) = 5
Is the method intelligent^ It is at least as intelligent as
any other method, and it has this advantage that the boy
works his sums exactly as he works ordinary simple mul
tiplication. And the procedure is easily and immediately
justified, by conversion and reconversion.
Give other examples, using the same numbers but changing
the position of the decimal points. The answers shall be
given mentally and at once:
72314 X 32 = 2314048.
0072314 X 32 = 02314048.
72314 X 00032 = 2314048.
72314 X 320 = 72314 X 32
= 2314048
If preferred, the boy might set out his working thus:
72314 x 32 = ^4 x ^
2 31 4 4 R
~iTyfrofy<r
= 2314048
and show his actual multiplication neatly on the left.
Possible objections to the method.
1. " The most important digit in the multiplier is not
used first." Granted. But this disadvantage is outweighed
by the advantage of greater accuracy.
2. " The decimal points are not kept in a vertical column."
This is of no material consequence, though it is quite easy
to teach the boy, if it is thought worth while, to place the
points in the successive products. For instance:
72314
32
144628
216942
2314048
84 CRAFTSMANSHIP IN MATHEMATICS
The boy multiplies through by 2 and then says, " When I
multiplied 4 by 2, I multiplied thousandths by hundredths,
and this gives me hundredths of thousandths, which occupy
the 5th decimal place; therefore the point goes in front of
the first 4." He argues similarly when he has multiplied by
the 3, though he would soon learn that the position of the
point in the first partial product gives the key to its position
in all the other products. Thenceforth he would work
mechanically. Does not the time come when we all work
mechanically in all types of calculation? does not the rationale
of procedure tend to fade away, until something turns up
demanding revivification?
Is there a more intelligent plan than teaching the boy to
complete the actual multiplying before considering the
decimal point at all? I doubt it. And I am quite sure that
no other plan is productive of a greater degree of accuracy.
The boy has confidence in a method so closely akin to one
with which he is already familiar.
Standard Form
It has been gravely said that " standard form " was the
invention of the devil. In reality it was not quite so bad as
that. It was invented* by an old personal friend of my own,
the senior mathematical master of one of our great Public
Schools, who decided that he " must adopt some new method
to prevent his boys from getting so many sums right, in order
to take the conceit out of them ".
Why are the apologists of the method always so faint
hearted?
* The method was suggested by '* standard " form in logarithms, where of
course it is very useful.
DECIMAL FRACTIONS 85
Division
As division is the reverse process of multiplication, the
analogous method for fixing the decimal point may be adopted.
2314048 f 32
_ 2 :i 1 4 O 4 N
_ 2 .'I 1 4 4 H
 a a
= 72314 x
__ 7 2 :) 1 4
 1 006"
= 72314.
The actual simple division by 32 may be neatly shown to the
left.
The simple rules are:
1. Ignore the decimal points and divide as in simple
division.
2. Subtract the number of decimal places in the divisor
from the number in the dividend. This gives the number
in the quotient. Fix the point by counting back this number
from the right.
The whole process resolves itself into (1) conversion,
(2) division by whole numbers, (3) reconversion.
Again: is the method intelligent? Again the answer is that
it is as least as intelligent as any other method, and it certainly
makes for accuracy. Here is an example with the working
as commonly shown: Divide 20735 by 872.
_ 23
872)20735
1 744
~3295 Decimal places = 4 2 = 2.
2616 Thus the quotient is "23, and a remainder.
1379
The division may now be continued to any number of places.
If, before dividing, we add O's to the dividend and
continue the dividing further, this does not affect the
86 CRAFTSMANSHIP IN MATHEMATICS
decimal point in the quotient: e.g. divide 2073500 by
872.
2377
872)2 : 073500
1 744
3295
2616
6790 Decimal places =62 = 4.
6104 Thus the quotient is '2377, and a remainder.
~6860
6104
~756
Hence, if a given dividend contains a smaller number of
decimal places than the divisor, add O's to make the number
equal (and more if necessary). Example: divide 001 by
70564. Write:
70564>OOl
We cannot proceed with the division until we add at least
5 more O's.
1
70564y00100000
70564 Decimal places = 84=4
29436
Thus the answer is '0001 . . . and a remainder. The quotient
can be carried to as many places as may be required.
The value of the remainder. It is desirable to make
the abler boys see the real value of the quantities in the
successive steps of the division. Example: divide '07925
by 37.
214
37)07925
74
62
37 Decimal places = 51 = 4.
155 Quotient = '0214 and a remainder.
148
7
DECIMAL FRACTIONS 87
What is the value of the " 74 " in the first step? It is the
product of 37 and 02 (as we may see from the quotient),
and must therefore contain (1 + 2 or) 3 decimal places;
hence its value is '074. Similar arguments apply to each
step. Hence, more correctly, the division may be set out in
this way:
0214
37JM37925
074 = 37 X 02
0052
0037 = 37 X 001
00155
00148 = 37 X 0004
00007
Thus the quotient (to 4 figures) is '0214, and the remainder
is *00007. The abler boys will soon learn to assign the
correct value to the remainder, by merely glancing at the
dividend vertically above it.
Verification should be encouraged:
Dividend = (quotient X divisor) + remainder
= (0214 X 37) + 00007
= 07918 + 00007
= 07925
Practice in manipulation of the following kind is useful:
3204 = 3204 = 03204 = 3204
^0701 ~~ ~7(M ~ 000701 ~ 701 '
The boy sees at once that the same quotient must result
from all the division sums. The only real defence for the
reduction of the divisor to a form approximating a small
whole number is that it enables a boy to obtain a rough
answer by easy calculation. For instance, in the last of the
r r 1. 3204 320 , . , ,
four forms above, =^r = =~ approximately, and thus the
88 CRAFTSMANSHIP IN MATHEMATICS
answer to this group of division sums is roughly y of 320,
i.e. a number between 40 and 50.
. '. 000983 0983 983
Again: ^ = or _ .
Thus the answer is roughly ~ 4 ~ 7  of 98 hundredths, i.e. about
2 hundredths, i.e. about 02.
This is useful for final verification, but the decimal point
is best fixed by the rule already given. If the simple multi
plication and division are accurately performed, the correct
fixing of the decimal point is a simple matter to even an
unintelligent boy.
Recurring Decimals
These will probably rarely be used, except in a very
simple form. Every boy ought, however, to know their
significance, though as a subject of general exposition they
are now generally ignored, perhaps unwisely. The younger
race of mathematicians are losing familiarity with much
that is interesting in the theory of numbers. Most people
know of course that \  142857142857 . . .= 14285?, and
that if we multiply this group of 6 figures by 2, 3, 4, 5, and
6, respectively, we obtain products giving the same group
of figures in the same order, each succeeding group be
ginning with the next higher figure of the group. It is,
however, less commonly known nowadays that this re
markable property of numbers is not uniquely characteristic
of the sevenths but applies to all prime numbers whatsoever,
7 and beyond, and that the grouping within the groups is
sometimes of an extraordinarily interesting character. Teachers
of arithmetic probably lose not a little of the potential effective
ness and interest in this subject if they do not familiarize
their pupils with some of the properties of numbers, properties
which to beginners seem almost uncanny. (See Chapter XLI.)
As to circulating decimals, a boy should be taught at
least this much:
DECIMAL FRACTIONS 89
Show him that he may at any point bring to an end the
quotient of a decimal he is dividing, showing the remainder
as a vulgar fraction.
Thus I = 14; or 1428 J or 142852.
Reconvert, say, the first: 14=  ^ + ^ = 1^1 = '<><> = *.
Then the boy sees that the scheme is justified.
Again: J = 3333 . . . apparently without end,
so, J = 1111 . . . apparently without end,
and, J,J 717171 . . . apparently without end.
But we can bring the division to an end anywhere, e.g.:
i = 3333J, or 33J or 3.
1 = 111?, orlj,.
H =7171JJ or71JJ.
Reconvert these:
ol
.01 _ ; __ 1 O __ :i O _ 3
;i "" io~" ;l ~ '*
1 1
.11 . 1< i i _ 10 _ 1
'' ~ ~ lltf ~ "'
10
.71?! _ 7 _i x i) it _ 7 i 170 _ 7100 _ 71
'  1 '. ( .) 10 ' TA i i" u y o i> o o '.) TT
171
x i) it _
Now show the repeated figures in a decimal division this way:
3333... = 3; 1111... = 1; 717171... = tl.
Thus we have learnt that
Hence to convert any repeating decimal into a vulgar fraction,
we make a denominator of 9's, viz. as many as there are places
in the decimal.
Thus Si = fJ; '69 = =
90 CRAFTSMANSHIP IN MATHEMATICS
If there are nonrepeating figures as well as repeating figures,
e.g. 5l6, then
1A 6ia_6j!!_ 811 .
.516 = _  _  ,
.01 A l ' 6  l "  i  i
016  loo " 100 ~ TW ~ **'
The commoner forms should be known, especially the thirds,
sixths, and twelfths:
I = .16; = 83.
^ = 083; ^ = 416; /^ = 583; }4 = 916.
The boys should know that when the denominator of a
vulgar fraction contains any prime numbers except 2 and 5,
the conversion of the fraction to a decimal is bound to give
a circulating decimal.
Simplification of Vulgar and Decimal Fractions
There are certain conventional rules about signs; for
instance:
1. Multiplication and division must be given precedence
over addition and subtraction:
3X 18 +15 f3 2 means (3 X 18) + (15 ~ 3)  2.
2. Multiplication and division alone must be worked in the
natural sequence from left to right:
36 r 9 X 2 means (36 ~ 9) X 2.
But the conventions are not wholeheartedly accepted; they
are without reason, and they are traps for the unwary. It is
unjust for examiners to assume that they will be followed.
The above examples should have been written with the
brackets. If brackets are inserted mistakes need not arise.
Here is a complex fraction to be simplified, taken from
one of the best textbooks in use. Doubtless the question
DECIMAL FRACTIONS 91
was taken from an examination paper. If so, the examiner
should have been put in the stocks.
1463 jj of 14175  j X 88125
7315 5 X 189 + 25 + 122
If given at all, it should have been written
^ 3 + f (lon^M^^
7315 I (5 X 189) + 25 f 122
It is often an advantage to work in decimals instead of
vulgar fractions. Example: What fraction of 21, 5$. 6d.
. 04255 X 32 f ri , ,
1S .  of /I, 11$. 3rf.?
00016 * '
. 00016
rraction =
04255 x 32 ,. OK ,. f
of 3125 shill.
4255 shill.
04255 x 32 x 3125
^00016 X 4255
1 2
x & x 3125
x
1 1000
= 625
At the third step, both numerator and denominator were
multiplied by 10 9 , to get rid of the decimals. Boys feel more
confidence when cancelling whole numbers. But in A Sets
such conversion should not be necessary.
Decimalization of Money
The common method of performing arithmetical opera
tions on money, weights, and measures reduced to their lowest
denomination has the advantage of simplicity but the dis
advantage of tediousness and cumbrousness. It is certainly
an advantage to work in the highest denomination when
possible, decimalizing all the lower denominations. For
92 CRAFTSMANSHIP IN MATHEMATICS
instance, if we have to multiply 432, 175. \d. by 562, it is
obviously an advantage to multiply 43286875 instead,
provided we can convert into the decimal form at once.
But the rules for conversion to more than 3 places are a
little too difficult for slower boys, and it must be remembered
that if multiplication is in question (and this is often the
case), exact decimalization is necessary, or the multiplied
error may be too serious to be negligible. The mil invariably
causes trouble. On the whole, decimalization methods are
advisable in A Sets, not in others.
But all boys should be taught to give in pounds the
approximately equivalent decimals of sums of money, i.e.
to call every 2/, !; every odd I/, 05; every /6, 025;
every farthing, 001.
Thus: 3, 17s. lOJrf.
 3 + 16/ + I/ + /G + IS/.
= (3 f 8 + '05 f 025  018)
= 3893.
Greater certainty and greater accuracy is obtained by the
ordinary method:
\d.  &d.
= 875.9.
175. lOJrf.  178755.
= 89375
3, 17s. 10JJ.  389375,
the boys dividing by 12 and 20 without putting down the
factors.
The converse operation, the conversion of decimally
expressed money into pounds, shillings, and pence, is most
safely and quickly performed by the oldfashioned multiplica
tion method (by 20 and by 12).
Consider 30789275.
89275 = 17855s.
8555. = 1026rf.
.'. 30789275 == 307, 175. 10 26rf.
= 307, 17s. 10^. (+ Old.).
DECIMAL FRACTIONS 93
Numerous tests in recent years have shown conclusively
that the usual decimalization rules are productive of much
inaccuracy amongst slower boys. But decimalization and
reconversion by the ordinary methods of division and multi
plication are easy to effect and are often advantageous in
practice.
A quick boy who wanted to multiply 15s. 9rf. by 2420
would probably use the practice method (15s. = of 1,
9rf. = ~^ (  of 15s., and he would see at once that the product
is 1815 + 9075 or 1905, 15s.), and he would not
decimalize. But to a slow boy a choice of methods is only
an embarrassment. He wants one method, and that method
without frills of any kind.
Contracted Methods
Contracted methods of multiplying decimals are productive
of so much inaccuracy that their use with average boys is not
advised. In A Sets, of course; in B Sets, perhaps; in C and
D Sets, no; though in A and B Sets logarithms will usually
be used instead, unless the sum to be worked is so simple
that ordinary methods are quicker. To slower boys logarithms
are puzzling, and their use in lower Sets is not recommended.
But no boy ought to be allowed, in Forms above the Fourth,
to show in his working the figures to the right of and below
the heavy line in a sum like the following.
Divide 5286143 by 3729 (to 4 places).
1417
3X^)5286143
3729
1557 1
1491 6
65
37
82
54
29^
253
26 103
2 150
94 CRAFTSMANSHIP IN MATHEMATICS
I have found that even slower boys soon gain confidence in
cutting out figure after figure in the divisor, instead of
bringing down figures from the dividend. The doubtful
" carry " figures worry him a little at first, but not for long,
and he soon learns to understand what to do to ensure
accuracy to a given number of decimal places.
Never encourage average boys to adopt the expert mathe
matician's plan of multiplying and subtracting at the same
time (Italian method). Boys hate it, rarely become expert
at it, and make mistakes galore. It should of course be used
by boys having any sort of real mathematical bent.
CHAPTER XI
Powers and Roots. The A, B, C of
Logarithms
Powers and Roots
I have seen four and fivefigure logarithms deftly used in
Preparatory Schools, but it is probably not wise to expect
much facility before the age of 14. The A, B, C of logarithms,
as a simple extension of work on powers and roots, may,
however, readily be taught a little sooner.
At first, powers, indices, and roots should always be
treated arithmetically, not algebraically. The later general
izations are then much more likely to be understood.
Some typical preliminary exercises:
(i) 5 4 = 5 X 5 X 5 X 5; 5 3 = 5 X 5 X 5;
.. 6* X 6 3 = (5 X 5 X 5 X 5) X (5 X 5 X 5)
= (5x5x5x5x5x6x5)
= 5%
/. 6 x 5* = 5 7 .
POWERS, ROOTS, LOGARITHMS 95
Thus lead up to the rule, and then state it clearly, that
in multiplication of this kind the indices are added. But
impress on the boys that the operation concerns powers of
the same number (in this case, 5), though any number may be
similarly treated.
(ii) 7 9 = 7x7x7x7x7x7x7x7x7;
7 4 = 7x7x7x7;
\X\X\X\
79
L = 74 = 75
74
Now lead up to the associated rule of subtraction of indices,
in division of this kind.
(iii) 7 4 = 7X7X7X7;
7 4 _7x7x7x7__,
" 7 4 "" 7~X 7 X 7~x~7 ~ '
. 744 ^ i or 70 = i So 30 ^ 1; 10 o = j^
(iv) (7 2 ) 3  7 2 X 7 2 X 7 2
 (7 X 7 X 7 X 7 X 7 X 7) = 7 6 .
Thus lead up to the rule as to multiplication of indices.
State categorically that such results always hold good, and
that a convenient way of remembering them is this:
But at this stage do not talk about " general laws ". Let
the above expressions be looked on merely as a kind of short
hand for collecting up several results actually worked out
arithmetically.
The following is a summary of a particularly effective
first lesson I once heard, given to a class of boys of 13, on
fractional and negative indices.
" The square root of a number is that number which
when squared produces the original number; e.g. the square
root of 16 is 4; of 81 is 9. We write R16 = 4; R81 = 9.
96 CRAFTSMANSHIP IN MATHEMATICS
(The mathematician writes his R like this: ' v /> an d calls
it ' root '. Thus * V 36 = 6 ' reads ' root 36 is 6 ')
carefully that V 6 X V 6 = V^ = V 62 = 6 ; that
X y'll = 11; and so on.
" The cube root of a number is that number which when
cubed produces the original number. We show the operation
by writing a little 3 inside the <\A Thus 4/125 = 4/5 X 5 X 5
 4/5 3  5; 4/1000  4/10 X 10 X 10~= 4/10 3  10;
4/5 X 4/5 X 4/5 = 4/12.5 = 5. _
" And so on. 4/81  4/3~X 3x3x3  4/3 1 = 3.
" Now suppose the index is a fraction, and not a whole
number. What does 5* mean? Well, we have learnt that
52 x 5 2 = 5 2 + 2 = 5 4 , so apparently we may assume that
5* X 5*  5*+* = 5 1  5.
" But V 5 X V 5 = 5; therefore 5* = V 5 
In other words, 5* is merely another way of writing down \/5.
Similarly, 5* X 5* X 5*  5* + * f *  5 1  5; therefore
5* = 4/5, i.e. 5* is another way of writing down 4/5.
" Similarly, 4/7  7 1 ; 4/2  2*.
" What does 8* mean? We know that (3 4 ) 3 = 3 12 , so
apparently we may assume that 8* = (8 2 )*, i.e. that 8 J means
the cube root of 8 2 , or 4/8 2 , or 4/64, or 4.
" Similarly 5 3 = 4/5 3  4/125.
" Thus we have learnt that the numerator of a fractional
index indicates a power ', and that the denominator indicates
a roo.
" Again, what does 6~ 2 mean?
" Since 6 5 x 6 2 = 6 5+2 = 6 7 , apparently we may assume
that 6 5 X 6~ 2  6* 2  6 3 .
But & f 6 2 = ^  6 3 ;
.'. 6^ X 6~ 2 
or
POWERS, ROOTS, LOGARITHMS 97
Similarly, 5 3 = I ; also 1 = 7~ 4 .
Thus we may conveniently remember that a~ n and are
two ways of writing down the same thing.
" Examples:
7 i _ _
75
"
A root form in a denominator is often troublesome,
since it leads to difficult arithmetic; and we may often get
rid of it in this way:
1= J_ X ^=^= W6"
A/5 V5 V5 5 "
The lesson was followed up by a few very easy exercises.
There was no algebra, save the " shorthand " expressions
utilized as mere mnemonics. The teacher's purpose was to
make the boys familiar with the basic facts of indices (integral
and fractional, + and ), and with the alternative forms of
writing down the same thing. Naturally many more examples
were given than the few above cited, and by the end of the
lesson the boys were remarkably accurate in their answers to
" mental " test exercises that were made somewhat severely
searching.
Boys should know their squares up to 20 2 . Extraction of
square roots may be taught when (a + b) 2 is known in algebra,
though boys should be made to break up numbers into
factors whenever possible, and then to obtain square roots
by inspection. Encourage boys to leave certain types of
answers in surd form, but, generally, to rationalize their
denominators; thus the answer  would not be acceptable,
V 7
(E291) 8
98
CRAFTSMANSHIP IN MATHEMATICS
but y 5\/7 would. All boys should know the values of
V'S, \/5, \/T, to 2 places of decimals.
The Beginnings of Logarithms
" Mathematicians were long ago clever enough to see
how they could use indices for working long sums in multi
plication and division. Suppose they wanted to multiply
together two large numbers, each of which was a power of 3.
They would turn to a book of * tables ' showing the powers
of 3. In fact, we may easily make up a little table for ourselves:
e.g. 3 l  3; 3 2 = 9; 3 3 = 27; 3 4  81; &c. Here is a
table from 3 1 to 3 16 . In the first column we write the index,
in the second the corresponding number.
Index.
Number.
Index.
Number.
1
1
9
19,683
2
9
10
59,049
3
27
11
177,147
4
81
12
531,441
5
243
13
1,594,323
6
729
14
4,782,969
7
2187
15
14,348,907
8
6561
16
43,046,721
" Now for some exercises.
1. Multiply 19,683 by 729.
Answer: 19,683 x 729 = 3 9 X 3 6 = 3 16 = 14,348,907.
2. Divide 43,046,721 by 531,441.
Answer:
531,441
3 12
3. What is the square of 2187?
Answer: (2187) 2  (3 7 ) 2 = 3 14
= 81.
4,782,969.
How easy! Instead of working hard sums, we simply refer
to our table, and add, subtract, or multiply little numbers like
9, 6, &c.
POWERS, ROOTS, LOGARITHMS 99
" But the mathematician would set out the first sum some
thing like this:
Index of the answer = index of (19,683 x 729).
= index of 19,683 + index of 729.
= 9 + 6.
= 15.
" But in the table the number corresponding to the index
15 is 14,348,907.
.'. Answer = 14,348,907.
Rather a roundabout way, isn't it? And he has a rather
grand word which he prefers to the word index, it is ' log
arithm '. The second part of the word, arithm, you already
know; the first part, log means * rule ' or ' plan '. Although
logarithms are only indices, the word itself suggests a clever
" arithmetical plan " for shortening our work, and you must
try to master it.
" Our little table contains only a few numbers, and there
are big gaps between them; e.g. there is no number between
27 and 81. Now 27  3 3 and 81 = 3 4 . Would it be possible
to obtain a number between 27 and 81 by finding the value
of 3 3 *? Certainly we should suspect that the value of 3 3 *
is somewhere between 27 and 81.
" We know that 3 3 *  3*  ^ = ^2187 = 468 (by cal
culation). Hence 3 3 * does lie between 3 3 (27) and 3 4 (=81).
Obviously it is possible to put into our table as many fractional
indices as we like, and so make the table more complete.
" The 3 which we have made the base of our calculations
the mathematician calls a base. Any other number might
be used instead, and in point of fact 10 is generally used."
The boy is now in a position to understand that (base) 100
= natural number, and he may be introduced to a short
table of threefigure logarithms, a table that may be included
in a single printed page. Give a variety of very easy examples,
and avoid great masses of figures. It is enough at this stage
to drive home the main principle. There is much to be said
100 CRAFTSMANSHIP IN MATHEMATICS
at first for avoiding the word logarithm altogether, and for
letting the boy work from the relation, number = lO indMf .
But we are anticipating Form IV work. (See Chap. XVII.)
CHAPTER XII
Ratio and Proportion
Simple Equations Again
If 4 chairs cost 20, what is the cost of 15 chairs?
4 chairs cost 20.
/20
.*. 1 chair costs .
4
^ i 20 X 15 ..,
/. 15 chairs cost ^ 75.
4
This method, " the method of unity ", is a good childish
way of working such a sum, and it is the method suitable
for boys up to the age of 11. At about this age the notion of
ratio should be introduced and it should gradually supersede
the unitary method.
First, revise the work on very simple equations.
We may begin a sum by saying, <k What number of
chairs . . . ?" The number we do not know; we have to
discover it. It is customary to let the symbol x represent a
number not yet discovered, and to argue about the x just as
we argue about any ordinary number.
How many chairs can I buy for 45, if 1 costs
Let the number of chairs be x.
Then 5 x x = 45,
/. 5* = 45.
/. * = Y = 9
Similarly, if 11 x = 51, x =  3.
RATIO AND PROPORTION 101
" We have already learnt that we may multiply or divide
the two sides of an equation by any number we please,
provided that we treat the two sides alike; e.g.
If x = 12, then 3x = 36, or ~ = and so on.
If we have an equation involving fractions, it is an advantage
to get rid of them as soon as we can, and we may always do
this by multiplying both sides of the equation by the L.C.M.
x 20
of the denominators: e.g. let the equation be  = . The
L.C.M. of 9 and 12 is 36. Multiplying both sides by 36,
we have
4x = 60.
/. x = 15.
Instead of using the L.C.M. for our multiplier, any other
C.M. will do, though this will mean rather harder arithmetic.
We might, for instance, use the product of the denominators,
viz. 108.
x = 20
9 12*
/. 12jc = 9 X 20.
x 15 (as before).
In this form we see in the simplified second line all four
terms of the original equation (x, 9, 20, 12), and this sim
plified second line might have been obtained at once from
the original equation by cross multiplying, i.e. by mul
tiplying each numerator by the opposite denominator.
This crossmultiplying is often very useful, in algebra and
geometry as well as in arithmetic.
" From crossmultiplying it follows that if we have two
equated fractions, a numerator and the opposite denominator
may be interchanged; e.g. = \ * , 2 7 8 = J .f , ^ 2 ~ =  2 \.
" Now we come to Ratio and Proportion."
102 CRAFTSMANSHIP IN MATHEMATICS
Ratio and Proportion
If 1 sheep cost 3, then,
2 sheep cost 6
3 9
4 12
5 16
7 sheep cost 21
10 30
13 39
21 63
As we increase the number of sheep we increase in the same
proportion the number of pounds.
Take any pair of numbers (sheep) from the first column,
and the corresponding pair of numbers (pounds) from the
second, say the last but two and the last in each case, and
convert them into fractions, thus:
153*
We see that these fractions are equal. That we should
expect, for 10 bears the same relation to 21 as 30 bears
to 63. A better way of saying it is that the ratio of 10 to 21
is equal to the ratio of 30 to 63.
" We know that the sign of division is ~, and that if in
the place of the two dots we write numbers, e.g. , we have
a fraction, and that the fraction means 5 divided by 6. Thus
a fraction represents a quotient. Similarly a ratio represents
a quotient. A ratio merely shows the relation between two
quantities, viz. how many times one is contained in the other.
When two ratios are equal, as in the case of the sheep and
pounds, we write them thus:
10 30
aT *3>
and we read,
10 bears the same ratio to 21 as 30 bears to 63.
Such a statement is a statement in proportion. Sometimes
we read ' 10 is to 21 as 30 is to 63 ', and sometimes * 10 over
21 equals 30 over 63 '.
" Remember, then: a statement in proportion is a
statement of the equality of two ratios.
RATIO AND PROPORTION 103
How many pounds of tea can I buy for 40$. if 6 Ib. cost 15s.?
Call the unknown number of pounds, x. We have 4: terms
viz. 2 lots of pounds, 2 lots of shillings. Write:
Lb. Shillings.
x cost 40.
6 15.
Convert each pair of terms into a ratio or fraction, equate, and
6 15"
.'. 5x = 80.
.'. x  16.
16 Ib. of tea cost 40s.; how many pounds can I buy for 155.?
Lb. Shillings.
16 cost 40.
x 15.
16 40
Equating, = .
x 15
.. 40# = 240.
/. x = 6.
Find the cost of 6 Ib. of tea if 16 Ib. cost 405.
Lb. Shillings.
6 cost x.
16 40.
Equating, 1 = i.
/. 2x = 30.
.. x  15.
6 /6. of tea cost 15$.; etfAtf* is the cost of 16 Ib?
Lb. Shillings.
6 cost 15.
16 x.
v 6 15
Equating,  = .
16 x
:. Qx = 240.
104 CRAFTSMANSHIP IN MATHEMATICS
The simple scheme applies to all cases of direct proportion
write down the 4 terms in pairs; equate; solve for x.
I sell a horse for 47, 10$., thereby losing 5 per cent. What
should I have sold him for if I had gained 5 per cent?
n . . ^ , Representative
Prices tn pounds percentages.
47J 95
x 105
47J 95
Equating,  =
= 52,
95 2
The prices of. the horse are in direct proportion to the re
presentative percentage numbers.
Inverse Proportion
But, of course, inverse proportion is another story. In
practice it is relatively rare, and is thus sometimes overlooked.
To cover a floor with carpet 72 in. wide I require 40
yd. from the roll; if the carpet is only half the width, I require
twice the number of yards from the roll; if only onethird
of the width, then 3 times the number of yards. We may
tabulate thus:
Running yards. Width in inches.
40 72
80 36
160 18
320 9
Clearly we cannot select a pair of terms from one column and
equate them to the corresponding pair from the other. One
pair has to be inverted, e.g.
Thus when one quantity varies inversely as another, the in
RATIO AND PROPORTION 105
version of one ratio (it matters not which) is necessary before
equating.
Teach the boys to distinguish between direct and inverse
proportion by asking themselves whether when one quantity
increases the other increases or decreases, and to distinguish
them on paper by pointing arrows in the same direction to
indicate direct proportion, in opposite directions to indicate
inverse proportion.
3 Ib. of tea cost 8s.; how many Ib. will cost 24$.?
Lb. Shillings.
1 3 8 i
* x 24 *
6 men can do a piece of work in 20 days\ how many could
do it in 15 days?
Men. Days.
1 6 t 2
* x I 15
Boyle's Law is the best known example of inverse proportion
in science, but in practical life examples of inverse proportion
are much less common than those of direct, and the conse
quence is that very artificial examples are often invented to
illustrate it. " Men and work " sums are often silly. " If
it takes 20 men to build a house in 20 days ", more than
one maker of an arithmetic book will ask us to believe that,
as a logical consequence, 1000 men could build the house in
I of one day.
Never use the old form of proportional statement, : :: : .
A common (and meaningless) form of statement sometimes
found in a boy's exercise book is
36 : 40 :: 24.
Always let the equated ratios consist of two fractions, and
make the boy realize that the particular position of the x
(first, second, third, or fourth place) is entirely without
significance.
106 CRAFTSMANSHIP IN MATHEMATICS
Examples Acceptable and Unacceptable
We give two more examples.
1. A clock which was 1$ minutes fast at 10.45 p.m. on
2nd December was 8 minutes slow at 9 a.m. on 1th December.
When was it exactly right?
This problem, like most other problems, requires a
preliminary discussion. By judicious questioning, help the
boys to cast it in a simpler form:
A slowgoing clock loses 9^ of its false minutes in 106J true
hours. In how many hours will it lose 1$ of its false minutes?
" Minutes " lost by slow clock during Hours of true clock
; ; 106i i
iJ x
9J ^ lOGJ
ij *T*
85 _ 425
" 13 " 4*"*
JL = _^
" 13 ~ 4i"
/. x = 161,
i.e. the slow clock was right 16J hours after 10.45 p.m. on
2nd December, i.e. at 3 p.m. on 3rd December.
2. It takes 8 men 6 days to mow a field of grass. How
long would it take 20 men to do it?
Days Men
f
8
20
6 = 20
x ~~ 8"
x = 2f .
But although 2 days is the orthodox answer, the time would
really be rather less. Men mowing a field for 6 days would
RATIO AND PROPORTION 107
find, in the growing season, the work much harder on for
example, the sixth day than on the first, so that the amount
of grass cut would not be equally distributed over the 6 days.
The answer as calculated is but a rough approximation.
Writers of textbooks, and some examiners, are so often out
of touch with practical life that it may be useful to append
a few absurd questions of the type supposed to be examples
of Ratio and Proportion:
1. It takes 3 minutes to boil 5 eggs. How long would it
take to boil 6 eggs?
2. A man rides a bicycle at the rate of 20 miles an hour.
How far could he travel in 92 hours?
3. My salary is 500 a year and I save 50 a year. How
long shall I take to save 10,000?
4. My brother weighed 24 Ib. when he was 3 years old.
How much will he weigh when he is 45 years old?
5. A rope stretches  in. when loaded with 1 cwt. How
much will it stretch when loaded with 10 tons?
6. It cost 1 to dig and line a well 2 ft. deep. How
much will it cost to dig and line a well 100 ft. deep?
7. A stone dropped down an empty well 16 ft. deep
reaches the bottom in 1 second. What is the depth of another
well, if a stone takes 5 seconds to reach the bottom?
Another point: if the answer to problems concerning men
and work comes out to, say 4J men; instruct the boy to say
5 men, with an explanatory note.
Until a boy is thoroughly well grounded in Ratio and
Proportion, the formal statement of the ratios is desirable.
But at least the abler boys in the top Forms may be allowed
to do as mathematicians themselves do write down the
odd term and multiply at once by the fraction determined
by the ratio of the other two terms.
3 1 Ib. of tea cost llf shillings; find the cost of 1\ Ib.
Cost = llfs. x f .
3 s
108 CRAFTSMANSHIP IN MATHEMATICS
" Compound " Proportion
" Double" or " Compound " "Rule of Three ".
For the most part the typical sums given by the textbooks
to illustrate this " rule " have little relation to practical life.
Occasionally they are legitimate enough, and then they may
be regarded as just a simple extension of the simpler two
ratio examples already considered. The terms may be
arranged in their natural pairs, converted into ratios, these
marked (with arrows) direct or inverse, then multiplied out.
If 16 cwt. are carried 63 miles for 6, 6$., what weight
can be carried 112 miles for 2, 16s.?
cost in
cwt. miles. shillings
\ x A 112 I 56
* 16 I 63 * 126
x
112 126
= 4.
Here is another, one of the commoner types, taken from one
of the best of the textbooks: If 36 men working 8 hours a
day for 16 days can dig a trench 72 yd. long, 18 ft. wide, 12/J.
deep, in how many days can 32 men working 12 hours a day
dig a trench 64 yd. long, 27 ft. wide, and 18 ft. deep?
The example is not practicable. Men working 12 hours
a day can not do 1J times as much work as men working
8 hours a day. The cost of digging a trench 18 ft. deep is
more than \\ times the cost of digging one 12 ft. deep. The
deeper the trench the more expensive it is to get out the
excavated earth. The cost does not necessarily vary as the
width of the trench; if timbering the sidfcs is necessary (a
serious additional item of expenditure), a little extra width
would not add appreciably to the cost. But more than this:
for excavation work, steam navvies have largely replaced
manual labour.
So it is with a large number of the textbook exercises:
RATIO AND PROPORTION 109
they have no relation to practical life. Here is one more,
from a really excellent textbook. If 10 cannon which fire
3 rounds in 5 minutes kill 270 men in 1 hours, how many cannon
which fire 5 rounds in 6 minutes will kill 500 men in 1 hour?
Did the man who made up this problem claim to be a mathe
matician, or a soldier, or a humorist? It is a shocking thing
that school boys are made to waste their time over the pretence
of " solving " problems of this kind.
CHAPTER XIII
Commercial Arithmetic
No branch of arithmetic is more important, and yet it
need not take up a very great deal of time. For the most
part, the work consists of the application of principles, already
learnt, to business relations in practical life. Once the boy
grasps the inner nature of the business relation, the arith
metic should give him little trouble. But " hard " sums,
especially sums involving great masses of figures, are rarely
if ever necessary. Give ample practice in working easy
exercises and so make the boy thoroughly familiar with the
A B C of commercial life.
Percentages
Teach the meaning of " per cent " thoroughly. We
require a numerical standard of reference of some kind, and
the number 100 has been accepted as the most convenient,
though any other would do instead. It is a disadvantage
that 100 is not divisible by 3.
5 per cent means 5 per 100 or 1 jfo. Drive this cardinal
fact well home: everything hangs upon it. 5 per cent of
110 CRAFTSMANSHIP IN MATHEMATICS
1 = y^ of 20*. = 1*.; 2J per cent of 1 = 6df. Let these
two results be the pegs of plenty of mental arithmetic; e.g.
1\ per cent of 1 = 3 times 6rf. = Is. 6d., and 7 per cent
of 20 = 1$. 6d. X 20 = 30^. ; and so on.
Representative percentage numbers (as they are usefully
called) is the next thing to drive home. When we buy a
thing, it may be assumed that we buy it at the standard
price which is represented by 100. If we sell the thing at
10 per cent profit, we sell it at a price represented by 110;
if we sell it at a loss of 15 per cent, we sell it at a price
represented by the number 85. This notion is of fundamental
importance. The majority of exercises grouped under the
term " percentages " or " profit and loss " are cases of
simple proportion, the two terms of one ratio consisting of
money and the two terms of the other ratio consisting of
representative percentage numbers.
How much is 12f per cent of 566, 13*. 4rf.?
Direct proportion example (common): If a debt after a
deduction of 3 per cent becomes 210, 3s. 4rf., what would it
have become if a deduction of 4 per cent had been made?
Reduced debts.
97 , 210*
96 * x
13 16
i i
= 208.
Inverse proportion example (comparatively rare): A
fruiterer buys shilling baskets of cherries, 30 in a basket. He
also sells them at a shilling a basket^ but 24 in a basket. What
profit per cent does he make?
COMMERCIAL ARITHMETIC 111
The smaller the number of cherries he sells in a basket,
the larger his profit. Hence the proportion is inverse.
Cherries per
Basket.
Representative
% Nos.
1 30
* 24
A 100
' X
x =
"
= 125.
This representative percentage number shows that the profit
is 25 per cent. (Strictly, the answer is not right, as no allow
ance is made for the necessary purchase of new baskets.)
Another inverse proportion example: If eggs are bought
at 21 for Is., how many must be sold for a guinea, to give a
profit of 12 per cent?
The selling price is represented by 112, a number
greater than 100; the number of eggs sold for a guinea must
be smaller than the number bought for a guinea. Hence
the proportion is inverse.
Ar r 01 / Representative
No. for 21/
441 I 100
x * 112J
. 441 X^
112J
=441 X f
= 392.
Simple Interest
The kind of examples really necessary should cause
little trouble. Even a slow boy readily understands the main
principles. As soon as he has learnt what 5 per cent per
annum means, he can follow this reasoning:
112 CRAFTSMANSHIP IN MATHEMATICS
Interest on 100 at 5 per cent per annum for 1 year
Interest on seven times 100, i.e. on 700 for 1 year
= 700 X Ttld
Interest on 700 for 3 years = 700 X T o X 3.
There is now an excellent opportunity for establishing a
simple algebraic formula:
Let I = Interest.
P = Principal.
R = Rate per cent.
,, T = Time in years.
Then I PxT J*R^ PTR
men i 1Q()  10Q .
The technical term " amount " should also be explained:
AP + I.
As interest is usually paid halfyearly, " 5 per cent per
annum " (as in the case of Government Stock) generally
represents rather more than its nominal value. This should
be explained.
The use of the formula is quite legitimate, provided the
boy has learnt to establish it from first principles; and equally
he may be allowed to deduce the subsidiary formulae, arguing
in this way:
Since from first principles
i _ PTR
loo'
/. I X 100 = PTR.
P = L2L1? . T  L* I?? R = * x 1QO
" ~ TR ' PR ; PT "
But in practical life these subsidiary problems (to find P or
T or R) are very rarely wanted, and it is not worth while
to let boys waste time over working a large number. An
occasional example, mainly to give facility in the use of the
formula, is enough.
COMMERCIAL ARITHMETIC 113
Compound Interest
It is enough to tell a boy to find out what will be due to
him if he places in the Bank 100 on deposit and allows
it to remain there for 2 or 3 years, the interest, say at " 4
per cent ", being undrawn. Two minutes of explanation
will show him how to work the sum, each halfyear's interest
being added to the Principal as it becomes due. A little
later on, instruction will be necessary as to shorter procedure
in calculation, but to give up time to the working of numerous
examples is inadvisable. Bankers never work compound
interest sums: they merely refer to readymade tables,
prepared by mathematical hacks for all the world to use.
Do not let the boys waste time over such useless work,
especially as the time is so badly wanted for other things.
On the other hand, see that they really do understand main
principles, and can readily apply them to simple cases.
The subject may, of course, be resumed in the Fifth or
Sixth Form, should the general mathematical theory of
interest and annuities be taken up.
Present Worth and Discount
Here again the principles are important and are very
easily mastered. Their use may be amply illustrated by
reference to a few easy examples. Do not forget to give a
clear explanation of Bills of Exchange and Promissory
Notes.
The boy already knows from his interest sums that
Amount =? Principal + Interest.
In Discount sums, three new terms are used, and really they
are identical with the three just mentioned:
Sum Due = Present Worth + Discount.
(1291) 9
114
CRAFTSMANSHIP IN MATHEMATICS
Here are two exactly analogous examples in direct proportion
of the normal type.
What is the Principal that
will produce an Amount of 840
in 3 years at 4 per cent?
When we have found the
Principal, we can subtract it from
the Amount, and so obtain the
Interest.
Principal,
x
Amount.
840
What is the Present Value of
840, the Sum Due at the end
of 3 years, the interest being 4
per cent?
When we have found the
Present Value, we can subtract
it from the Sum Due, and so
obtain the Discount.
Present Value.
x
Sum Due.
840
But there is only one term for each of the Ratios. Where are
the others? We have to invent them.
We do not know the value of x, the Principal that will
amount to 840 in 3 years at 4 per cent.
We do not know the value of x, the Present Value of the
sum 840 due in 3 years at 4 per cent.
But we may take any sum we please and invest it for 3 years
at 4 per cent. 100 is as good a sum as any.
100 invested for 3 years at 4 per cent yields 12 interest.
Thus 100 is the Principal that Amounts to 112 in
3 years at 4 per cent, and 100 is the Present Value of 112,
the sum due in 3 years at 4 per cent. Now we may complete
our Ratios.
Principals. Amounts.
* 100 ^ 112
x
ioo
840
If2'
15
= 750 (Principal) .
Interest (if required)
 840  750 = 90.
Sums Due.
I 840
* 112
Present Values.
4x
100
x __ 840
100 112*
15
/. * = 100 x g
2
= 750 (Present Value).
Discount (if required)
= 840  750 = 90.
COMMERCIAL ARITHMETIC 115
Remind the boy that this True discount is never heard of
in practice. The Bill Broker's discount, which he deducts,
is really the interest on the whole Sum Due. It is exactly the
same as calculating Interest on the Amount instead of on the
Principal, a thing the banker would (naturally) never dream
of doing. Let the boy compare the two things, and see for
himself that when the Banker deducts interest on the Sum
Due instead of on the Present Value, the customer receives,
as Present Value, a sum rather less than by arithmetic he
is entitled to.
Exercises in Present Value and Discount are hardly
worth doing, unless they are very simple and can be done
quickly.
Stocks and Shares
Nothing is more important in arithmetic than a working
knowledge of stocks and shares and of financial operations.
Whatever views political extremists may take of a roseate
financial future, we have to deal with the hard facts of the
present day, when it behoves every member of society to
save, and to invest his savings.
But do not make boys waste their time by working through
the useless examples on stocks and shares given in many of
the older textbooks.
The first stile for the boy to get over is the distinction
between stock and money, and there is no better way than
to turn the whole class into an imaginary Limited Liability
Company with its own Directors. To play a game of this
kind is worth while. Let the Directors draw up a simple
Prospectus and invite subscriptions at par. A week later
let the Directors report some disaster perhaps the destruc
tion of property by fire and an inevitable fall in the expected
interest. Some shareholders will become anxious and will
be willing to sell at 90 or even lower. And so on. A little
reality of this kind is worth ten times the value of a long
sermon on the subject. If a boy pays 100 (any sort of paper
116 CRAFTSMANSHIP IN MATHEMATICS
token will do) and receives a Certificate for 100 1 shares,
and then has to part with his shares at, say, 18$. each, a
sense of reality is brought home to him. Perplexity about
stocks and shares is almost always due to a hazy understanding
of the reality which underlies it all. As always, the trouble
is with the slower boys. The quicker boys pick up the threads
readily enough.
There are numerous facts for the boys to understand and
remember, as well as sums to work. Explain the nature of
debentures, preference shares, the different kinds of ordinary
shares, their relative value and relative safety. Warn the
boys never to invest without taking advice, and never in any
circumstances to invest in a new flotation. Explain " gilt
edged " securities, and point out the relative safety of Govern
ment stock, though even this may fall seriously in value
(compare the present price of Consols with the price fifty
years ago). Insist that a large interest connotes a big risk,
that financial greed spells disaster. Impress upon the boys
that the financial world is full of sharks.
The old days of a brokerage of ^ per cent have passed
away, and thus many of the sums in the older text
books are out of date. Stockbrokers' charges now include
Government Stamp Duty, Company's Registration fee,
and Contract Stamp. Give the class a short table of
charges to be entered in their notebooks, for permanent
reference, e.g.
Purchases 50+ to 75, total charges 18s. 3d.
Purchases 75+ to 100, total charges 1, 3s. 3rf.,
and so on.
All ordinary " examples " in stocks and shares are in
stances of simple proportion (nearly always direct: there is
little point in puzzling young boys with the rule " the amount
of stock held is inversely proportional to the price "), and
they call for no comment.
Examples on the purchase and on the sale of small amounts
of stock and small numbers of shares are the only exercises
COMMERCIAL ARITHMETIC 117
that need be given. Let the exercises be typical of those
that in practical life the average man engages in.
Other Commercial Work
Rates and Taxes. A simple explanation of and a variety
of exercises in these are of great importance. Explain the
increase in both rates and taxes during the last few years.
Distinguish carefully between expenditure by the Govern
ment and expenditure by Local Authorities, and show why
both kinds of expenditure are inevitable. Explain how taxes
are imposed and how rates are levied. Let exercises be easy,
but devise them to illustrate principles and to give an inner
meaning to things. " Rateable values " is another thing to
be explained.
And there are numerous other things, of which it behoves
every intelligent person to have at least an elementary know
ledge, things which only a mathematical teacher can handle
effectively. We mention a few: Income Tax and its assess
ment, its schedules, its forms and the correct method of
filling them up; rent, house purchase, mortgages; the raising
of loans by public bodies and by private persons; insurance
of all kinds, especially life insurance, Health and Unemploy
ment insurance; policies (especially " allin " policies) and
premiums; pensions, annuities, the keeping of personal
accounts, thrift, household economics; banks and saving;
the Post Office bank and National Saving Certificates; co
operative stores and their financial basis; building societies;
insurance tables and how to read them (a Sixth Form ought
to have some knowledge of their actuarial basis). There
are numerous tables of very useful kinds in Whitaker that
every boy ought to be made to understand, and by means of
them an arithmetic teacher may devise exercises of a very
valuable kind.
A particularly useful syllabus on the arithmetic of
citizenship is given in the appendix of the 1928 Report of
118 CRAFTSMANSHIP IN MATHEMATICS
the Girls' Schools Subcommittee of the Mathematical
Association.
Books on arithmetic to consult:
1. The Psychology of Arithmetic, Thorndike.
2. Lecons d'Arithme'tique, Tannery (Armand Colin).
3. The Teaching of the Essentials of Arithmetic, Ballard.
4. The Tutorial Arithmetic, Workman.
5. The Groundwork of Arithmetic, Punnett.
6. The Small Investor, Parkinson.
CHAPTER XIV
Mensuration
Simple Formulae
Easy problems involving actual measurements will be
embodied in the mathematical course for children below the
age of 11, by which time^ boy^ught to be familiar with the
mensuration of rectangular areas and rectangular solids and
to be able to work easy conversion (reduction) sums in linear,
square, and cubic measures. He ought also to have learnt
to measure up the area of the classroom floor and walls, and
to express his results in formula fashion) e.g. area of floor
= / X b\ area of the four walls = 2(1 + b)h
\He should now be taught, if he has not been taught
before, to make paper models of cubes and cuboids, and
from a consideration of the " developed " surfaces of these,
laid out in the form of " nets ", to devise formulae for cal
culating the areas;) e.g. of a cube, 6/ 2 ; of a square prism,
4/z/ + 2a 2 or 2a(2l + a)\ of a brick, 2(lb + It + bt). The
memorizing of these formulae is not worth while, but they
are worth working out as generalizations from particular
examples; and when, once more, numerical values are
assigned to them, it makes early algebra very real.j)
MENSURATION 119
The Papering of Rooms
Some attention must be given to the stock problems
on the papering of rooms, but it is not worth while to take
time over measuring up doors, windows, and fireplaces;
assume that the walls are unbroken planes, and the room
rectangular. Nor is it worth while to divide the perimeter
of the room by 21 in. to find the necessary number of strips
of paper. There is bound to be a good deal of paper wasted,
especially if the pattern is elaborate. Hence it is enough to
take the total wall area 2(1 ~\~ b)h, divide this by the area of
one roll of paper, 36 ft. X If ft. or 63 sq. ft. or 9 sq. yd.,
and so obtain the necessary number of rolls. If the answer
comes out to 13 rolls, evidently 14 are wanted, perhaps
15 because of waste; perhaps 13 or even 12 would do, because
of windows, doors, &c. A paperhanger never measures up
a room with any degree of accuracy; his estimate is very
rough and always done by rule of thumb. There is really
no point in giving boys such problems to work, especially
when it is remembered what a large number of problems,
depending on accurate measurements, may be culled from
the boys' physics course.
The Carpeting of Floors
The carpeting of floors is generally considered to give
an easier type of problem than the papering of walls, but
the problem in practice is a little tricky. If from an ordinary
27in. wide roll a carpet has to be made up to fit the usual
rectangular room, it is unlikely that the width of the room
is an exact multiple of 27 in., in which case the last of the
strips cut off the roll will be too wide, and there will be
waste; and yet the whole of that strip will have to be pur
chased, as the pattern cannot be " matched ". If the carpet
is plain, and the purchaser does not object to patching,
then the exact amount required may be cut from the roll,
120 CRAFTSMANSHIP IN MATHEMATICS
though the vendor might not agree to cut to the small fraction
of a yard.
Consider a floor 18' X 12', and carpet 2' 3" wide.
1. Let the carpet be plain (patternless). Area of floor
= 24 sq. yd. Required number of running yards from the
roll 24 r f = 32. This will give 5 strips, each 6 yd.
long, and 2 running yards (a piece 6' X 2' 3") over. This
strip of 6' X 2' 3" will have to be cut up to cover a space
18' by I)", so that it will be cut into 3 pieces each 6' long,
placed end to end, the width of these being 9".
2. Let the carpet show a design, the width being the same
as before. Evidently at least 6 strips, each 6 yd. long, or
36 yd. in all, must be cut from the roll. It is highly im
probable that, if the strips are cut to exact length, they would
match when laid side by side. There would be a good deal
of waste, depending on the size of the design. The problem
cannot be brought within the scope of classroom arithmetic:
all the factors are not available.
3. A more practical problem for the classroom is to
estimate the amount of plain carpet required to cover a room
of given size with a minimum number of complete strips,
allowing the surplus width to determine an equal allround
border (to be stained or covered with linoleum). For instance,
the 5 strips above mentioned would leave a surplus width of 9".
If the strips are placed together centrally, there will be a width
of 4" to spare at each end of the room. Hence we must
arrange for a complete border of 4" all round the room. Thus
the 5 strips will not now be 18' long, but 17' 3" long, and the
amount to be cut from the roll will be 17' 3" by 5 (28 yd.).
Thus the area of the room = 18' X 12'; of the carpet,
17' 3" X 11' 3"; and the required number of running yards
from the 2' 3" wide roll  28.
, If the whole floor had been covered, and patching was
allowable, the number of running yards required = 32; if
patching was not allowable, the number of running yards = 36
(leaving a waste piece 6 yd. long and 18" wide, with an area
of 3 sq. yd., equal to 4 running yards).
MENSURATION 121
Does it not all come round to this that these mensuration
problems concerning wallpaper and carpet are rather futile,
especially when whole chapters in arithmetic books are
devoted to them? Children are much better employed in
mensuration problems that really do enter into the practical
business of life.
Border Areas
Make these a matter of subtraction, whenever possible
as, for instance, in estimating the area of a garden path
4' wide between a rectangular lawn and the rectangular
garden wall the garden being 108' X 72'.
Area = {(108 X 72)  (100 X 64)} sq. ft.
Do not allow boys to find the area of the path piecemeal.
Rectangular Solids
For the mensuration of these, boys can, with very little
help, establish the necessary simple formulae and interpret
them in some brief form of words easily remembered. Prob
lems on the excavation of trenches, the cubical content of
cisterns, the air space of school dormitories, and the like,
will readily occur to the teacher. The cubical content of a
solid " shell " (e.g. of iron in a cistern, of stone in a rectangular
trough) should, whenever possible, be made a problem of
subtraction. Example: Find the weight of a stone trough
6" thick, external dimensions 10' X 3' X 2' 6", the weight of
stone being \\ cwt. to 1 cubic foot.
No. of c. ft. of stone = (10 X 3 X 2J)  (9 X 2 X 2)
= 7536
= 39.
Weight = 1J cwt. X 39 = 2 tons, 18 cwt.
A boy should never attempt to cube up the stone piecemeal.
If a gasholder (" gasometer ") at an ordinary gasworks is
made the subject of a mensuration problem, remember that
122 CRAFTSMANSHIP IN MATHEMATICS
(1) a gasholder has no bottom, (2) its top is not flat. Not all
writers of arithmetic books seem to realize this.
Mensuration beyond the very elementary stage is best
associated, primarily, with the geometry rather than with the
arithmetic.
CHAPTER XV
The Beginnings of Algebra
Informal Beginnings
Regarded as simple generalized arithmetic, algebra will
have been begun at the age of about 9 or 10. Quite young
boys will have measured up rectangular areas and will have
learnt to express intelligently the meaning of the formula
A = / X b. In their lessons on physical measurements,
rather older boys will probably have evaluated 77, 2?7R,
?rR 2 ; in their arithmetic lessons they will have established
PTR
the formula I = ; in their first lessons on Ratio and
x 1
Proportion, they will have learnt the significance of  =
6 14
and will have obtained the first notions of an equation.
Formally, algebra will not have been begun; informally,
foundations will have been laid.
Never begin the teaching of the subject according to the
sequence of the older textbooks. The difficult examples in
mechanical work so often given on the first four rules, on
H.C.F.s and L.CJVI.s, on fractions, &c., are not only calculated
to make boys hate the subject but are wholly unprofitable
either at an early stage or later.
Suppose you are asking questions in mental arithmetic
to a class of boys 10 or 11 years of age, and you suddenly
spring upon them the sum, " add together all the numbers
THE BEGINNINGS OF ALGEBRA 123
from 1 to 100." " We cannot do it, sir/'" Well, let us
try. Let us first take an easier sum of the same kind: add
together all the numbers from 1 to 12. We will do it in this
way:
" Add together the first number and the last, 1 and
12? 13.
" Add the 2nd from the beginning and the 2nd from
the end, 2 and 11? 13.
" Add the 3rd from the beginning and the 3rd from
the end, 3 and 10? 13.
" 4 and 9? 13; 5 and 8? 13; 6 and 7? 13.
Now we have included them all. How many 13's? 6. What
aresix!3's? 78. This 78 must be the answer to the question. " {
Smiles of agreement.
" Now let us make up a little formula that we can use
for similar sums: How did we obtain the first 13? We added
together the first number and the last.
" What is the first letter of the word first? f.
" What is the first letter of the word last? I.
" How can we show the sum of/ and /? / + /.
" How far along the line from 1 and 12 was our mul
tiplier, 6? Halfway.
" What is the first letter of the word half? h.
" Now I will show you how to write down/ + / multiplied
by h" Then follows a brief explanation, and A(/+ 0
" Now let us work the harder sum, 1 to 100.
"/= ? 1; "/= ? 100; "A = ?" 50.
" .'. h(f + /) = 50(1 + 100) = (50 X 101) = 5050.
Now add together all the numbers from 1 to 1000." And
so on. " I have been giving you an algebra lesson which I
sometimes give to boys 2 or 3 years older. An interesting
subject, isn't it?" Yes. " And useful?" Yes.
Or we might begin straight away with problems producing
equations. First notions of an equation will already have
124 CRAFTSMANSHIP IN MATHEMATICS
been given in arithmetic. By means of a few easy exercises,
revise the principle that the two sides of an equation may
be added to, diminished, multiplied, or divided, by any
number we please, provided that the two sides are treated
exactly alike. The rule of cross multiplication should also
be revised. But naturally at thi$ stage no equation should
be given with a binomial in a denominator.
Here are two examples, in a teacher's own phraseology
(summarized, except that, to save space, his many admirably
framed questions are omitted), once taken with a class of
beginners of 11.
The sum of 50 is to be divided among 2 men, 3 women,
and 4 boys, so that each man shall have twice as much as each
woman and each woman 3 times as much as each boy. Required
the share of each.
" In sums of this kind it is always well to consider first
the person who is to have the least, in this case a boy.
Let x be the number of 's in each boy's share.
Then 3x is the number of 's in each woman's share.
And Qx is the number of 's in each man's share.
Hence we have, in 's,
The share of the 4 boys = 4#.
The share of the 3 women = 9x.
The share of the 2 men = I2x.
But the sum of all these shares amounts to 50.
.'. 4* f 9* + 12* = 50,
25*  50,
/ want to divide some nuts among a certain number of
boys. If I give 4 nuts to each boy, I shall have 2 nuts to spare;
if I give 3 to each boy, I shall have 8 to spare. How many boys
are there?
" Let x be the number of boys.
" There are two parts to this problem, both beginning
with the word if\ each part enables us to write down the
THE BEGINNINGS OF ALGEBRA 125
total number of nuts> though not exactly as in arithmetic,
because we have to use x.
" 1. If I give 4 nuts to each of x boys, I give away 4#
nuts. But the total number of nuts is 2 more than that.
.'. the total number of nuts = 4# f 2.
" 2. If I give 3 nuts to each of x boys, I give away 3x
nuts. But the total number of nuts is 8 more than that.
/. the total number of tints  3# f 8.
.'. 4x + 2 = 3jc + 8,
.'. * + 2 = 8,
This class of boys subsequently spent the next three or four
lessons in working through a chapter of problems (some of them
pretty difficult) producing equations. They gained confidence
quickly, and from the first looked upon their new subject as
interesting and useful.
Formal Beginnings. Signs as "Direction Posts"
Of course the time comes when algebra must be treated
formally. There are certain fundamental difficulties that have
to be faced, and, of these, algebraic subtraction is to beginners
a difficulty of a serious kind.
This does not mean that boys need be taken through the
elaborate subtraction sums of the older textbooks. It means
that they have to be taught the inner meaning of, say, " take
3# from 2# ". To the boy, what does that meant At
first it can mean nothing but juggling with arithmetical
values, juggling of which he is naturally suspicious.
Consider, first, what the boy has already done (or ought
to have done) in his arithmetic. He is familiar with this kind
of sum. 17 _ 6 __ 5 + 3 _ 9 + 14
= 17 + 3 + 14659
= (17 + 3 + 14)  (6 + 5 + 9)
= 3420
= 14.
126 CRAFTSMANSHIP IN MATHEMATICS
He has been taught to collect up his numbers in this way,
and to realize that the plan of adding together all the minus
numbers and taking them away "in a lump" is a much better
plan than taking them away separately. Thus he sees that
34659
must be the same as
34  (6 f 5 + 9),
and he therefore gets a first notion of the effect of a minus sign
before a pair of brackets. Still, the work so far is wholly
numerical, and nothing more. If the sum had been
20 34
he would probably have been taught to prefix a + sign to
the 20:
+ 20  34,
to take the difference between the 20 and the 34, and to prefix
the sign of the larger number (34); thus, 14. Naturally
the boy would call it a " subtraction " sum, and would say
that the 14 is 'Mess than nothing "I The teacher might,
however, allow the use of this illogical expression provisionally,
comparing it with debts as against assets.
But the boy must soon come to grips with the funda
mental algebraic notion of direction, as well as of numerical
value.
In arithmetic always, in algebra commonly, the + sig n
before an initial term is omitted. But in the early stages of
algebra it is advisable that it be consistently written.
In algebra it is necessary to have some means of distin
guishing direction to the right from the opposite direction
to the lefty and direction upwards from the opposite direction
downwards. The opposed signs + and are used for this
purpose. These signs are the algebraic signposts or direction
posts; the two signs direct the numbers to which they are
attached. It has been agreed that direction upwards and to
the right shall be called a + (positive) direction, and direction
THE BEGINNINGS OF ALGEBRA
127
downwards and to the left a (negative) direction. The
converse would have done equally well, but the decision has
been universally accepted. It is just a convention. If the
boy asks why? tell him there is no answer.
Consider the centigrade thermometer, with the freezing
point marked 0. If the temperature is 5 and rises 20,
every boy knows that it rises to + 15; i.e. 5 + 20 = +15;
also that if the temperature is, say, +10, and falls 25,
it falls to 15, i.e. +10 25 15. The results may be
obtained by actual counting, upwards or downwards, on the
scale. Upward counting means adding + numbers; down
ward counting means adding numbers. The thermometer
provides an excellent means of giving a first lesson on algebraic
direction.
Addition and Subtraction
Now consider a more general case. We will choose a
horizontal scale, with + numbers and numbers to the
right and left, respectively, of a zero.
Adding + quantities means counting to the right.
Adding  quantities means counting to the left.
__ . ___  j pj
10 9 8 7 6 5 4 3 2 1
I I 1 I I I I I I I
+1 +2 +3 44 +5 +6 +7 +8 +9 +10
Four addition sums:
Addition Sums.
Starting Point
on Scale.
Count or
Add on Scale.
Direction,
Ror L.
New Point on
Scale and /. Ans.
i
(+5) + (+3)
+3
+5
R
+ 8
ii
( + 5) + (3)
3
+ 5
R
+2
iii
(5) + (+3)
+3
5
L
2
iv
(5) + (3)
3
5
L
8
Four subtraction sums. Where we have to work a
subtraction sum, say 12 7, we may work it by asking what
128
CRAFTSMANSHIP IN MATHEMATICS
we must add to 7 to make 12. Thus in the four subtraction
sums below we may say,
(i) What must we add to (+3) to make (+5)?
(ii) ,, ,, (3) (+5)?
(Hi) ,, (+3) (5)?
(iv) (3) (5)?
Subtraction Sums.
Starting Point
on Scale.
Scale Point
to count to.
Direction,
R or L.
Number of Points
counted
= Answer.
i
( + 5)  ( i 3)
+ 3
15
R
+ 2
ii
( + 5)  (3)
3
15
R
+ 8
iii
(5)  ( + 3)
+ 3
5
L
8
iv
(5)  (3)
3
5
L
2
i
Let the boys now examine the two groups of answers and
note from them that:
( + 5) + ( + 3) 
( + 5) + (3) =
(5) + ( + 3) =
(6) + (3) =
( + 6) (3)
( + 5) (43)
(5) (3)
(5) (43)
They thus learn that in every case we can turn a subtraction
sum into an addition sum merely by changing the sign of the
subtrahend.
They ought now to understand that in arithmetical addition
the total is increased by each term added; that in algebraic
addition the numbers indicate movements or distances back
wards or forwards along a line from a zero point or " origin ".
They ought also to see that in algebraic addition we may
drop the sign which separates the components, and deal
with the components in accordance with their own signs, e.g.
(45) + (43) 4543
(45) 4 (3) = +53
(5) + ( + 3) 5 + 3
(_5) + (3)= 53
For algebraic subtraction, let them substitute algebraic
THE BEGINNINGS OF ALGEBRA 129
addition, at the same time always reversing the sign of the
second term (subtrahend). Since the sum is now an addition
sum, we may drop the connecting + sign as before:
( + 5)  ( + 3) = ( + 5) + (~3) = +5  3
( + 5)  (3) = ( + 5) 4 ( + 3) 4513
(5)  (43)  (5) 4 (3) = 5  3
(5)  (3)  (5) 4 (43) 643
Examples: Add 11 x and 19*:
417*  19* = 2*.
From 17* take 19*:
417*  (19*)  17* 4 19* = 436*.
For the slow boys, indeed for all boys, the whole process
crystallizes into three simple little rules:
1. Addition sums.
(i) Like signs: add, and prefix the same sign,
(ii) Unlike signs: find the difference between the two
numbers and prefix the sign of the larger.
2. Subtraction sums. Reverse the sign of the second term
(subtrahend) and treat the sum as an addition sum.
Teachers are not always quite happy about this question of
directed numbers, and often ask if it is not unwise even to
make the attempt to deal with it, and if a statement of just the
rules ought not to suffice. Of the answer I have no doubt.
Boys who do not grasp the significance of directed numbers
can never get to the bottom of their algebra; their work
all through will inevitably be mechanical. Admittedly,
however, the nonmathematical boy fails to understand, and
for him the rules, as rules, must suffice. A Sets can and must
master the difficulty, and I think B Sets too. But with C Sets,
and especially with D Sets, be content to state the rules
and to give the boys plenty of practice in them. Such boys
will never make mathematicians, and nobody expects that they
will. It is best to admit that the application of signs to com
ponent and resultant scale distances is a little too subtle
for the nonmathematical boy.
(E291) 10
130
CRAFTSMANSHIP IN MATHEMATICS
M
+200
5 4 3 2
+2 +3
+5
Multiplication
Here again the rule of signs can be understood only by
a clear grasp of the effect of direction. The usual train
illustration is as good as any.*
Graph the route of a train travelling northwards through O
(say Oxford) at the rate of 40 miles an hour, and thus show
the position of the train at all points on its journey.
Let horizontal lengths to the right of MOM' measure
times after train reaches O, and let the times be indicated
by f numbers; and
let those to the left
of MOM' measure
times before train
reaches O, and let
these times be indi
_ H cated by numbers.
Let lengths above
H'OH measure dis
tances north of O,
and let these be indi
cated by + numbers;
and let those below
H'OH measure dis
tances south of O,
and let these be indicated by numbers.
We will mark the positions of the train 4 hours before
reaching O and 4 hours after passing O. (The scales used
are 5 mm. to 50 miles and 5 mm. to 1 hour.) At 40 miles
an hour, the train must, at these times, be 160 miles short of
O and 160 beyond O, respectively. Plot points P 2 and P l
to show this. P 2 must be directly below 4 on H'H, and to
the left of 160 on M'M; P l must be directly above +4
on H'H and to the right of +160 on MM'. The line P 2 P a
evidently passes through O, and represents the train route.
P 2 (I60)
60
ISO
200
M'
Fig. 14
See Nunn, Teaching of Algebra, Chap. XVIII.
THE BEGINNINGS OF ALGEBRA
131
Now how can we determine the two positions by cal
culation?
We may utilize the formula d = vt (" distance = speed
X time "), and by making the three symbols stand for
directed numbers, the formula will give us information about
the direction as well as the magnitude. Hence we must use
the term velocity. Let velocity northwards (40 miles an hour)
be considered f.
1. Position of train at P 2 :
d = vt = ( + 40) x (4)
= 160 (as graphed) = 160 miles S.
2. Position of train at P x :
d = vt = ( + 40) x ( + 4)
= +160 (as graphed) = 160 miles N.
Now consider the train travelling southwards. Let velocity
southwards (40 miles an hour) be considered negative ( ).
3. Position of train at P 3 :
d = vt = (40) X (4)
= +160 (as graphed) =160 miles N.
4. Position of train at P 4 :
d = vt = ( 40) x ( + 4)
= 160 (as graphed) = 160 miles S.
132 CRAFTSMANSHIP IN MATHEMATICS
Comparing the 4 results we have:
( + 40) x ( + 4) = +160
; + 40) x (4) = 160
(40) X (4) == +160
(40) X ( + 4) = 160
This is enough, at this stage, to justify the sign rule for mul
tiplication. A more rigorous generalization may, if necessary,
come later.
(The boys should be made to see that the sloping lines in
the above graphs do not graphically show the actual railway
track, which is supposed to run due northsouth.)
It may be urged that the whole thing seems to be a
little artificial. So it is. But the rule of signs is a universally
accepted convention. The convention is perfectly self
consistent, and is easily justified, but by its nature it admits
of no " proof ".
Book to consult: The Teaching of Algebra, Nunn.
CHAPTER XVI
Algebra: Early Links with Arithmetic
and Geometry
Algebra and Arithmetic in Parallel
Get the boys to see that an algebraic fraction is only a
shorthand description of actual arithmetical fractions, and
that there is really no difference in the treatment. The working
processes are practically identical.
The arithmetical fraction ^ may be written ?,
which shows clearly that the denominator is greater by 3
than the numerator. So does the fraction ^, and that
is all it means. Thus in the fraction ~j, a + 7 represents
ALGEBRA: EARLY LINKS
133
a single number; as in arithmetic, it must be moved as a
whole from one place in the expression to another. In algebra,
beginners sometimes forget this, and treat the parts of a
binomial denominator separately. So with 2 or more bi
nomial denominators: for instance in  r>4"jirfi> x ^
and x + 11 express single numbers.
Show a few corresponding arithmetical algebraic processes
side by side. It helps the slower boys much.
1. Let a = 4, 6 = 7.
28
L 1
28"
28

a b
b .,. 
ttb ab
2. Let a = 3, b = 4, c = 5
1 +  4 +  5
20 15 12
= 3 4 2 6*
~~ 60 60 60
32 4. 42 + 52
3. Let a = 7, b 4.
_1 J^ i<
7~ 4 7~+~i ' 49~~
= 1 1 _ 14
~ 3 11 33
 I 1 4 3 _ U
33 33 33
 H + 3 ~ 14
"33"
= JO
"" 33
 0.
be
4 4 ^
ca a6
6^ ^
^c a^
1 , 1
2a
a b a \~ b a 2 b 2
4 __1_ _ ___
a 2  b* a*  b*
^a b 2a
0.
134
CRAFTSMANSHIP IN MATHEMATICS
There is little or no need to take fractions beyond quite simple
binomial denominators. Denominators of a higher order are
seldom required in practice. Hence all H.C.F.s and L.C.M.s
should be evaluated by factorizing, exactly as in arithmetic.
The principle of the cumbrous divisional processes for finding
factors should be familiar to boys in A Sets, who, however,
may be referred to their textbooks. Do not waste time over
such things in class.
Geometrical Illustrations
Factors, multiplication, division, simple expansions, &c.,
should all, in the early stages, be illustrated geometrically,
and thus be given a reality. When this reality is appreciated,
but not before, the illustrations may be given up. Second
power expressions should be consistently associated with areas.
We append a few illustrative examples.
2
20
20
X
4
\ 20 * 4
4*
2
2 ^
at
> ab
b*
Fig. 1 6
1. Compare the square of 24 (i.e. 20 + 4) with the square
of a + b.
24 a (a + 6) a
= (20 + 4) 8 =fl 2 + 2a& + 6 2 .
= 20 2 f 2.20.4 + 4 2 .
2. Compare 24x27, i.e. (20 + 4) (20 + 7), with
ALGEBRA: EARLY LINKS
135
(20 + 4) (20 + 7)
= 20 a + 20.7 + 20.4 + 4.7
= 20 2 + 20(7 + 4) + 4.7.
(a + 4) (a + 7)
= a 2 + 7a + 4a + 4.7
= a 2 + a(7 + 4) + 4.7.
20
20 2
20
X
4
20*7
4
X
7
2
CL
I
a
X
4
r
\
a x 7
4
x
7
Fig. 17
3. Show graphically that
(2a + 56) (a + 36) = 2a 2 + Uab + 156 2 .
o. a. b b b b b
a
b
b
b
Fig. 1 8
The result is seen at a glance.
2
a.
2
a
ab
ab
.
ab
.
I
an
cxb
ab
b 2
b 2
b 2
b 2
b 2
db
ob
b ?
b 2
b 2
b 2
b 2
ab
ab
b 2
b 2
b 2
b 2
b 2
1
?<, + 5^
136 CRAFTSMANSHIP IN MATHEMATICS
4. Show graphically that (a 2) (a 3) = a 2 6a + 6.
A
* a 2 *
/
D
t
a. S
i
F
J
a
G
E
/
/
f
B
'* Q
C
AE = (a  2) (0  3),
AC  a 2 ,
GC  3a,
. 19
rcc\ l
== ^<z,
GC + FC  EC = 6fl  6,
AE  AC  (GF f FC  EC)
i.e. (a  2) (a  3) = a 2  5a f 6.
5. Show graphically that (a + 2) (a 3) a 2 a 6.
4
E / /
3
p
R
(a + 2) (a  3>
t
a3
P
K C
F
q
/ / / S
Fig. 20
(a + 2) (a  3) = AC I = AK + PF
= AK f EC I = AF  DQ  RS;
i.e. (a + 2) (a 3) == a 2 a 6.
ALGEBRA: EARLY LINKS
6. Show graphically that
(a + b + cf = a* + b* + c 2 + 2ab + 2ac + 2bc.
137
 a. *_b*4C*.
r
a
2
CL
ab
ac
5
ab
tf
be
[
I
f
ac
be
c 2
Fig. 21
The result is seen at a glance.
CHAPTER XVII
Graphs
From the Column to the Locus
Begin with column graphs, that is with mere verticals
with the tops unconnected; say, the amount of gas consumed
each week for a quarter, or the height of a barometer each
morning for a week. Now join the tops of the columns,
first by straight lines, then by a curved line. Do the straight
lines teach anything? Which is likely to be the more correct,
the straight lines or the curved line? Can intermediate columns
be inserted, and, if so, what would they signify? Are there
any cases where intermediate columns would be absurd?
Now discuss the locus graph, as distinct from the column
138
CRAFTSMANSHIP IN MATHEMATICS
graph. The barograph is a useful example especially for the
contrast of a gentle slope and a steep slope, and hence as an
introduction to a gradient and what it signifies. What does
a chart of closely packed isobars signify? of open isobars?
Or graph the vertical section of a piece of hilly country, by
taking heights from an ordnance survey map. Here the gentle
slope and the steep slope appeal at once, the closely packed
isobars and the closely packed contours being akin.
Dates (20
of 1^
lnnings[ MAY
The significance of a gentle gradient and of a steep
gradient is fundamental. It is really the key to all subse
quent work. Let the boys graph their cricket scores for
the previous summer term, and discuss the resulting
gradients. Familiar and personal data of this kind often
provoke animated discussion of a useful character. In the
first lesson or two, much of the work can be done on the
blackboard, exact numerical values playing only a minor
part. Give the beginners a general notion of the graph and
its significance. A few instances may be culled from chemistry
and physics, say solution curves (common salt, with its
GRAPHS
139
very slowly rising straight line; nitre, with its steep curve);
the experimental results, with their subsequent pictorial
illustrations, are always impressive. Other useful graphs
from practical work are a straightline graph from a loaded
spiral spring, or from a F.C. scale comparison; an inverse
proportion graph, say a timespeed curve during a journey.
Thus prepare the way for formal work.
15
penc
5
Ib
Fig. 23
A parcel post graph is easy of interpretation and, by
its gradient of equal steps, leads on naturally from a column
graph to a direct proportion graph. It may be called a
" stepped " graph. There is a minimum charge of 6d. for
any weight of parcel up to 2 Ib.; the charge is 9d. for any
weight over 2 Ib. and up to 5 Ib.; a shilling for any weight
over 5 Ib. and up to 8 Ib.; and so on. A straight line can
be drawn through the corners of the figure, but this straight
line does not pass through the origin.
The mere plotting of a graph nowadays gives little trouble.
Most modern books give instructions both simple and
satisfactory. But a clear understanding of what has been
done and a satisfactory interpretation of the completed
graph often leave much to be desired. It is the interpretation
that is the allimportant thing. A graph is essentially a kind
of picture, a picture to be understood. The pictorial element
admits of a general interpretation simple enough for be
140
CRAFTSMANSHIP IN MATHEMATICS
ginners to understand, but as time goes on this interpretation
must be made more and more exacting.
The study of y = tnx f c. Direct Proportion
Experience convinces me that the study of the form
y = mx should precede that of the form xy c. But pro
portionality of one kind or another underlies the whole thing,
and the straight line and rectangular hyperbola should occupy
a first place.
Do not attempt to define for beginners the term function.
The term should, however, be used from the first. " Here
is an expression involving x, that is, a function of x." In
time, drop the words " expression involving " and simply
say " function of ". Let the word be used constantly; it
will gradually sink in and become part of the boys' own
mathematical vocabulary.
Begin with a straightline graph passing through the origin.
(i) y = oc. What does this mean? That^y is always equal
to x, i.e. that the ordinate is always equal to the abscissa,
+3
+2
+I
3 2
O +1 +2
I
2
3
no matter what point on the line is taken, whether in the first
or third quadrant. Thus in the figure we have the point
(3, 3) in the first quadrant and the point (2, 2) in the
third.
GRAPHS
141
(ii)y == oc. This is practically the same as before. The
length of y is equal to the length of #, i.e. the length of the
ordinate is equal to the length of the abscissa, but now the
signs are different, whether a point is taken in the second
a;
Fig. 25
quadrant (as 3, 3) or in the fourth (as 2, 2). The graph
runs from the left downwards, from the second to the fourth
quadrant.
(iii) 3y = 2x. This means that three times the length of
tje ordinate is equal to twice the length of the abscissa. We
may write, more simply, y = \x, and then we see that the
(3,2)
Fig. 26
ordinate is always f of the abscissa. This is easily seen from
any pair of values (save 0, 0) in a table:
x = .3 _2 +3 44 4
y = _2 1
42 42}
142
CRAFTSMANSHIP IN MATHEMATICS
No matter what point in the line is chosen, the ratio of (1)
the L r to the x axis to (2) the intercept on the x axis, i.e.
the ratio ~, is always . This ratio is constant; the triangles
x
formed by drawing perpendiculars are all similar. The slope
of the line is always the same, i.e. the gradient of the graph
is constant.
(iv) 2y = 3x or y \x. Here the length of the
ordinate is always 1^ times the length of the abscissa, but
+ 3
+ 2
+ I
fl +2 +3
2 /
the two are of opposite signs, as may be seen from any pair
of values (save 0, 0).
x = 2 1 2 3 4
y = 3 1J 3 4J 6
The graph runs from the left downwards, from the second
to the fourth quadrant.
Before proceeding further, give the class plenty of mental
work from the squared blackboard, using a metre scale or
a rod to represent the graph, holding it in various positions
but always passing through some selected named point and
through the origin, and asking the class to name the equations.
GRAPHS 143
I have known a class of thirty boys give almost instant
response, one after the other, when tested in this way.
See that the boys become thoroughly familiar with the
difference between y = mx (same signs, slope from left
upwards) and y = mx (opposite signs, slope from left down
wards). Also see that they are not caught by the alternative
forms to these, viz. y mx = 0, y f mx = 0.
The next step is to see that the boys understand the sig
nificance of the wi in the equation y = mx. They already
know that when the coefficient of y is unity, the coefficient
of x is a ratio representing ?, i.e. the " steepness ", the
x
" slope ", or the " gradient " of the graph, and they are thus
prepared for the general method of writing this ratio, viz.
by the letter m. Do not begin with the general form m 9 and
say that it represents the slope of the line, and then illustrate
it with numerical examples. Begin with the numerical
examples, in order that the boys may really understand the
principle; then introduce the m as a sort of shorthand registra
tion of facts which they already know.
The next step is to move the graph about parallel to
itself, and to study the effect upon the written function;
and so lead the boys to see that a graph which does not pass
through the origin necessarily cuts off pieces (intercepts)
from both axes (we neglect the case of a graph parallel to
an axis). We may begin by graphing a few particular cases
of the function y = mx > say y f # c:
y = %x + 2
y = 1* + 1
y = t*
y = f*  1
y = f*  2.
Show the pupils how to tabulate two or three pairs of values
of each case, and how then to draw the graphs. They may
then compare their results.
144
CRAFTSMANSHIP IN MATHEMATICS
They will readily discover that the +2, +1, 0, 1, 2,
represent merely the number of units the graph has been
raised or lowered (the third case,;y #, being an old friend).
The function proper, y = f#, is the same in all cases\ the
slope is constant; the five lines are parallel. A perpendicular
(ordinate) dropped from any point on the graph to the x axis
\oc+2
Fig. 28
shows a rightangled triangle similar to all other similarly
drawn triangles. In every case, the ratio of the sides round
the right angle is given by the m, the coefficient of x. The
number (the c) added or subtracted represents merely the
bit of the y axis intercepted between the graph and the x axis.
For this reason we call such bits of the y axis, intercepts.
But when we raise or lower the graph above or below the
origin, the graph really intercepts both axes. If the graph is
raised above the origin, a portion of the y axis above the
origin is intercepted, and a portion of the x axis to the left
GRAPHS
145
of the origin, as well. If the graph is lowered below the origin,
a portion of the y axis below the origin is intercepted, and
a portion of the, x axis to the right of the origin, as well.
How in each case are the two intercepts related?
Consider the first of the above five expressions, viz.
y = f # + 2. Instead of expressing y in terms of #, we may
express x in terms of y, thus:
v y = %x f 2
/. 3y = 2* f 6
Here the # intercept is 3, where we have precisely the
same graph as before when the y intercept was +2. The
2 '
Fig. 29
+ 1
Fig. 30
function is unaltered. So with the last of the five expressions,
viz. y = f# 2. If we express x in terms of jy, we have
x == 2^ + ^ The a? intercept is +3, and, as before, the y
intercept is 2, the graph being identically the same. The
function is unaltered, we have merely expressed it differently.
Generally, however, we express y in terms of x, and the
added or subtracted quantity (the c) represents a y intercept.
The analogous results from the function y = f x i c
may now be rapidly dealt with in the same way.
Let the pupils occasionally check a graph by means
of other pairs of tabulated values. For instance, from the
function y = f x + 2 we have:
(K291)
3 +3 +6 +8
+2 +4 +6 +7}
11
146
CRAFTSMANSHIP IN MATHEMATICS
Consider the last point (8, 7J), where OS = 8 and PS = 7J.
The slope of the graph is determined by the sides round
the right angle of any rightangled triangle determined
in the manner aforementioned. In the main figure we
see two such triangles (shown also as separate figures
Fig. 31
with the ordinates, x and j, in dark lines). The slope is
determined
PR ^ y _~ 2 = 7^ 2 = 2
RQ x 8 3'
 PS v 7* 2
or by
either by
TS 3 + x 3 + 8
 2
y _
Hence we may write either
The two are identical. * ' 6 ' 6 + x ' 6
Beginners are apt to confuse the value of m with the
coordinates of some arbitrarily chosen point; e.g. to take
the value (8, 7) of the above point P, to convert it into the
74
fraction ?, and to call it m. It is a thing that wants watching,
8
The boys ought now to realize that, in y = mx + c,
the c is of little consequence compared with the allimportant
m\ and that it may sometimes be convenient to ignore the
c and to plot the graph in its fundamental form y = mx.
Since it then passes through the origin, the function is more
easily recognizable.
The linear function should thus provide the boy with a
GRAPHS 147
preliminary training to enable him to see clearly how the
relation between variables may be represented not only in
equation form but pictorially. He should be able to discover
the relation between the variables, that is, to discover the
equation or law connecting them, to discover what function
y is of x, to discover m.
The beginner is often perplexed when told that Ax + By
+ C = is the general form of a linear equation. Why
those capital letters, he wonders. But if he first sees that
his now familiar friend y = f x + 47 ma y be written
3x + 4y = 18, he will understand that the new form provides
a neater way of writing down the function, though the all
important m no longer reveals itself so readily. " When
we write this new and neater form Ax f~ By + C = 0, the
only reason for using capital letters is that it enables us to
identify it readily. Other forms and their specific uses you
will learn all in good time. Why should we not have different
ways of writing down the same function? May we not weigh
up in the laboratory a piece of brass in ounces or in grams?
Convenience dictates a choice of method."
It is a good general plan to lead up to a general form
through a few particular examples. To spring suddenly
upon a class such a general form Ax + By + C = 0, before
they have been suitably prepared, is not the sort of thing
that an experienced teacher ever does.
Independent and dependent variables are terms to be
introduced gradually. Make quite clear that the x axis is
always used for the quantity which is under our control and
is quite " independent " of the other quantity, and that for
this reason it is given the name independent variable; and
that the y axis is used for values calculated from the formula,
or for values observed in experiment, i.e. values which
" depend " on the selected and controlled x values, and it is
therefore called the dependent variable. Each time we change
the value of our selected x quantity, calculation or observa
tion gives us a related y quantity; and the graph we draw is
a picture .to show not only how these pairs of quantities are
148
CRAFTSMANSHIP IN MATHEMATICS
related but to show that this relation is the same for every
pair.
Another way of expressing the connexion between the
two variables is to say that the dependent variable is a function
of the independent variable, the latter being often called the
argument of the function, since we make it the basis of our
argument. The graph of an equation shows how the function
varies as the argument varies and is called the graph of the
function; the abscissa is selected for the argument, and the
ordinate thus represents the function.
The Circle
There is little to gain in spending much time over the
circle, as it will rarely be used except to illustrate the solution
of such simultaneous equations as x 2 + y 2 = 52, xy = 24.
But it does serve to illustrate simply how a formula is affected
where the graph is " pushed about ". We give the same
circle in four different positions.
Fig. 33
Centre of circle at origin. Equation: x 2 + J> 2 = r 2 .
The centre is pushed 1 units to the right; its coordinates
are (1J, 0). The horizontal of the rightangled triangle is
no longer x, but x diminished by 1J.
Equation: (x 1 J) 8 + jy 2 = r 2 .
GRAPHS
149
The centre is pushed 2j units up; its coordinates are
(0, 2). The vertical of the rightangled triangle is no longer
jy, but y diminished by 2J.
Equation: x 2 + (y  21) 2 = r 2 .
Fig. 34
Fig. 35
The centre is pushed li units to the left and 2J units
up. The horizontal of the rightangled triangle is x + 1>
and the vertical is y 2.
Equation: (* + H) 2 + (y 2) 2  r 2 .
The Study of ;ry = ^. Inverse Proportion
The direct proportion graph we found to be a straight
line. The inverse proportion graph (the rectangular hyperbola)
is naturally the next for investigation.
Let the learner himself plot some simple case: " 32 men
take 1 day to mow the grass in the fields of a farm. How
many days would it take 16, 8, 4, and 2 men, and 1 man to
do it?" (An absurd example, really, but for our present
purpose the weather conditions and the growth of the grass
may be ignored.)
150
CRAFTSMANSHIP IN MATHEMATICS
With half the number of men, twice the number of
days would be required.
With one third the number of men, three times the
number of days would be required.
And so on. Hence, for graphing, we may write down these
pairs of values.
men \ 32
days  1
16
2
8
4
2
16
1
32
The graph is evidently a smooth curve. Lead the class to
discover that the product of each pair of values is constant,
that xy is 32 in all cases.
Fig. 36
Now plot xy = k for several values of k, e.g. k 25,
49, 64, 100, 225, 400, and examine the curves as a family.
How are they related?
1. A line bisecting the right angle at O divides all the
curves symmetrically.
2. The point where that line cuts the curve is the point
nearest the origin; it is the " head " or vertex of the curve.
GRAPHS
151
3. At a vertex V, x = y. Hence, . xy = k y x = y =
. . in *jy = 25, the coordinates of the vertex V are (5, 5).
4. Each curve approaches constantly nearer the axes, but
k
never reaches them. However great the length of x, y = 
and y can therefore never be zero. Neither can x ever be
zero. Either may be indefinitely small because the other
may be indefinitely large, but neither can be absolutely zero.
Hence we say that the axes are the asymptotes of the curve.
oc at  25
d
ecu 49
DC of * 64
ocu ;oo
DC
 225
400
/o
/5 20 25 30
Fig. 37
This term means that the line and the curve approach each
other more and more closely but never actually meet (asymptote
= " not falling together ").
5. The successive curves are really similar, although at
first they do not appear so. But draw any two straight lines
through the origin to cut the curves and examine the in
tercepted pieces of the curves (it is best to cover the parts
of the figure outside these lines), and each outer bit of curve
will be seen to be a photographic enlargement of the next
inner bit.
Boyle's Law is the commonest example of inverse pro
portion in physics. But the data (p and v) obtained from
school experiments are usually too few to produce more
than a small bit of curve, much too small for ready inter
152 CRAFTSMANSHIP IN MATHEMATICS
pretation. But inasmuch as the law pv k seems to be
suggested by the data, this may be verified in two ways:
(1) find the product of p and v for each pair of related values
and see if the product is constant; (2) convert the apparently
inverse proportion into a case of direct proportion by plotting
not v against p but  against p. The points thus obtained
v
ought to lie on a straight line, and the line may be tested by
means of a ruler, or a piece of stretched cotton. Does the
line pass through the origin? Why?
There is probably little advantage in teaching boys to
" push about " into new positions the rectangular hyperbola,
though for purposes of illustration one or two examples may
usefully be given. If the graph xy ~ 120, or y ^, is
raised, say, 3 units, the function becomes y = ^..o + 3 or
y 3 = ' jo. If it is i owere d 3 un i ts> y _j_ 3 ^ .1 .? o. If
it is raised 3 units and then moved 4 units to the right, the
function reads y 3 = i~ ( ] or y = ^ + 3. But the
beginner is apt to find this a little confusing. It is best to
let him keep the curve in a symmetrical position, and to
continue to use the asymptotes for his coordinate axis.
Negative values. Instruct the class to graph xy = 100
for both positive and negative values. Then proceed in this
way.
11 When we plotted pairs of quantities from a linear
function, we passed from negative values through the origin
to positive values (or vice versa), and the graph was con
tinuous an unbroken straight line. Apparently, then, the
rectangular hyperbola, though consisting of two separated
parts, ought to be regarded as a single continuous curve.
Is this possible?
" The curve in the third quadrant is certainly an exact
reproduction of that in the first.
" Suppose the x axis indefinitely extended both ways, and
a point Z far out to the right to travel along it towards O
the origin. At any position it may be regarded as the foot
of the ordinate of a corresponding point P on the curve.
GRAPHS
153
As (for instance) Z x moves to Z 2 , P x moves round the curve
to P 2 , and as ZO diminishes in length (ZjO to Z 2 O), the
ordinate PZ increases (P^ to P 2 Z 2 ). But however long
PZ may be, it gets still longer as Z gets still nearer O. In
fact, it seems to become endlessly long, and yet we cannot
say that the curve ever really meets the y axis, for it is absurd
3U
nc
Of)
\
XI
c
/ **
oo
15
V
PZ
ID
\
^
^^
Ff
r

'<
5 2

p /
5 /
3 :
> O
*5
r i
2
1
5 2
z , 2
J i
^
k
IO
\
^16
\
~9n
\
f 2S
\
3o
Fig. 38
to speak of the quotient $. But if Z continues its march,
it must eventually pass to the other side of O. And yet no
interval can be specified to the left and right of O so short
that there are no corresponding positions of P still nearer
to the y axis on the right at an endless height and on the
left at an endless depth. As Z proceeds along QJC, P simply
repeats in reverse order along the curve in the third quadrant
its previous adventure along the curve in the first. The
crossing of Z over the y axis at O seems to have taken P
154 CRAFTSMANSHIP IN MATHEMATICS
instantaneously from an endless northern position to an
endless southern position. We feel bound to regard the
two curves as two branches of the same graph, for both are
given by the function xy ~ k = 100.
" If you plot xy k, the branches appear in the second
and fourth quadrants."
The above argument is always appreciated by A Sets,
though naturally its implications are too difficult for them
to understand until later. With lower Sets, it is futile to
discuss the subject at all.
With A Sets, too, the use of the term " hyperbolic func
tion " is quite legitimate. We called ax + b a linear function
of x because the graph of y ax + b is a straight line.
Similarly we may call any function that may be thrown into
k
the form + b a hyperbolic function of x, because the
x ~\~ a
k
graph y = + b is a (rectangular) hyperbola.
x f a
The Study of y = x 2 . Parabolic Functions
The pupil should master two or three new principles
before he proceeds to the quadratic function.
1. The first is the nature of a "root" of a simple equation.
A very simple case will suffice to make the notion clear.
The boy knows already that the root of the equation x 3 =
is 3. Now let him graph the function y = x 3.
Since y x 3 we have:
* =

1
2
3
y = X 3 =
3
2
1
The line crosses the x axis at +3, that is when y = 0, x = 3,
and we say therefore that 3 is the " root " of the equation
# 3 = 0. Of course we should never let a boy waste his
time by actually solving an equation in this manner, but
it serves to teach him that when the value of a function
GRAPHS
155
equals 0, then the intercept on the x axis gives the root of
the equation represented by the function. (Fig. 39.)
The roots of related equations are easily derived. For
instance, solve the equation #3=1. (Fig. 40.)
'3.0 '
Fig. 39
Fig. 40
Write y = x  3 = 1; i.e. y = x  3, and y = 1. The
graph of y = x 3 is the same as before; the graph of y = 1
is a line parallel to the x axis, 1 unit above. The value of x
in the equation # 3 = 1 is given by the intercept that
y = x 3 makes with y = 1, i.e. 4. In other words the root
of the equation is the x value of the point of intersection of
the two lines.
Evidently we have the clue for solving graphically two
" simultaneous " equations, say, x2y=l, and 2*+3y=16.
The lines cross each other at the point P (5, 2). This pair of
values satisfies both equations (let class verify). A line drawn
through this point parallel to the x axis is y = 2. Hence
the value of x for both lines where they cross y = 2 is 5.
The 5 represents the intercept on the line y = 2, made by
each of the given lines. (Fig. 41.)
2. A second preliminary principle to be mastered concerns
the method of making out tables of values for graphing.
Having decided what values of x are to be used (this is a
question of experience), write them down in a row, then
156
CRAFTSMANSHIP IN MATHEMATICS
evaluate the successive parts of the function, one complete
row at a time. The mental work proceeds much more easily
this way than when columns are completed one at a time.
For the sake of comparison, we will set out selected values
12
Px
X 2i
2oc + 5i
L
Fig.
1
IG
of the function 4# 2 4# Je^, in two ways, one by addition,
one by multiplication. Show the learner why the results are
necessarily identical.
x=
o
36
12
2
16
8
U
1
4
4
i
1
2
+ 1
4
4
+ 2
+ 2i
+ 3
36
12
+4
64
16
33
i
4r 3 :
4z
9
6
16
8
25
10
y~4:X z 4:X~ 15  =
33
9
n
15
16
15
7
9
Since the function factorizes into (2x + 3) (2x 5), we
may set out the values of the factors and multiply, instead
of adding as before:
x~
3
2
li
1
i
+ 1
+
+ 2J
+ 3
+4
(2*+3) =
(2*5) =
3
11
1
g
o
+1
7
3
5
4
_4
5
3
7
1
8
9
1
11
3
y = 4#* 4# 1 5 =
33
9
7
16
16
16
7
9
33
GRAPHS
157
3. A third preliminary principle concerns scales. Different
scales for the two axes are often desirable, though in the early
stages of graphing different scales are not advisable. The
learner should recognize the normal slope of the straight
line and the normal shape of the curve. Only in this way
can he recognize and analyse the purely geometrical properties
of the graph. But with the study of the parabolic function,
if not before, the " spread " of the numbers should be taken
into account. Moreover, a good " spread " to the parabola
is an advantage, in order to obtain accurate readings of the
x intercepts.
We now come to the actual graphing of the function.
2,4
u4
2.4 C
2 o
2
 X
X
Fig. 42
Let the boy be first made familiar with the graph of
the normal function y x 2 , the parabola being head down
and the coordinates of its head (vertex) being (0, 0). Let
him see that the curve cuts any parallel to the x axis in two
points, e.g. 1 and 1, \/2 and \/2, &c. The curve is
symmetrical with respect to the y axis. Note that, with the
same scale for both axes, there is not much spread to the curve.
Now we will graph the function y 4# 2 4^15,
taking the sets of values for x and y from either of the tables
on the previous page. To obtain a greater " spread ", we
adopt a larger scale for the x axis. The curve cuts the x axis
(when . . y = 0) in two points, viz. 1 and 2 (these values
158
CRAFTSMANSHIP IN MATHEMATICS
are also seen in the tables), and these are therefore the roots
of the equation 4# 2 4#  15 0.
From the same graph we may obtain the roots of the
equations 4# 2 4x 15 = 9, or 4# 2 4# 15 7, or
4# 2 4# 15 = z y where z = any number whatsoever. It
is simply a question of drawing across the curve a parallel
V
3
\
33
\
2
ttLfi)
*
30
10
+ 2
/v
K33)
Fig. 43
to the # axis, and of reading the values of x from the points
of intersection. For instance, if 4# 2 4# 15 = 9, the
parallel to be drawn is x = 9, and this cuts the curve in
x = 2 and 3, which are therefore the roots of the equation.
These values of x may, of course, be seen from our tables
where y 4# 2 4# 15 = 9, but they are easily estimated
from the graph itself, if this is reasonably accurate.
A function may sometimes be conveniently divided into
two parts, and each part treated as a separate function and
GRAPHS
159
graphed. The intersection of the two graphs will then give
the roots of the equation. Really we have two simultaneous
equations; e.g.
if 4* a  4*  15 = 0,
then 4* 2 = 4* + 15.
Hence we may write
and
y = 4# 2
y = 4* + 15
The line cuts the curve at the points x = 1 and 2, and
these are the roots of the equation 4# 2 4# 15 = 0, as
10
**
*0
ocr
2 I
 4xM5
Fig. 44
before. It should be noticed that this last figure does not
represent the graph of the function y = 4# 2 4# 15,
though this graph is now easily drawn by superposing the
4* 2 graph on the 4* + 15 graph. If Y t = 4* 2 , Y 2 = 4^: + 15,
and Y = 4^ 2  4* 15, then Y = Y x  Y 2 . Hence any
ordinate of Y may be obtained by taking the algebraic difference
of the corresponding ordinates of Y! and Y 2 . Let the pupils
draw the Y graph from their Y x and Y 2 graphs, and verify.
160
CRAFTSMANSHIP IN MATHEMATICS
The function might have been broken up in another way
If 4x a  4*  15 = 0,
then 4# a 4* 15.
Hence we may write y = 4jc 2 4#
and y = 15.
(3,48)
\
/
K48)
\
/
\
,
y
\
rJ
\
(
'^'
\
i
b,2
>
\
If
= if
/
/
\
^
/
\
,l
tf
(
l,8>
\
^
/
'2,8)
^
15)
S
X
/
5210
(^
1) 1 2 3 4^
Fig. 45
Here are the graphs of these two functions. The latter cuts
the former at # 1^ and 2J, the same roots as before.
The easiest way to discover where the parabola 4# 2 4# 15
crosses the x axis is to express the quadratic function as a
product of two linear functions, viz. (2# + 3) (2# 5) = 0.
Hence either 2# + 3 = or 2# 5 = 0, i.e. x= f or ff.
Thus from the two linear functions we form two simple
equations, the roots of which are the roots of the quadratic
equation.
GRAPHS
161
We will plot these two linear functions (see the second
table, p. 156). (The lines happen to be parallel. Why?)
The graph of the quadratic function is readily obtained by
multiplying together corresponding y values (again refer to
second table, p. 156). For instance, at 2 the y value of
2x + 3 is 1 and the y value of 2x 5 is 9. The product
of 1 and 9 is +9. Hence at 2 the y value of the
quadratic function is +9, i.e. the point (2, +9) is a point
Fig. 46
on the curve. By pursuing this plan we may obtain fig. 43
over again.
The boy ought now to realize that he may graph his
function in a variety of ways. But do not encourage him to
think that the normal process of solving a quadratic equation
is to graph the function. Not at all. The important thing
for the boy to understand is that every algebraic function
can be thrown into a picture and that this picture tells a
story. What the algebra means to the geometry and what
the geometry means to the algebra are the things that matter.
We are dealing with the same thing, though in two different
ways, and the closeness of the relationship should be seen
clearly. As with the linear function, so with the parabolic
(B291) I*
162
CRAFTSMANSHIP IN MATHEMATICS
function: the boy must see the result of " pushing the graph
about ".
If
then
or
4x 15 = y,
2  4* + 1 = y + 16,
(2x  I) 2 = y + 16.
If we compare this with the normal form x 2 = y> we see that:
2x I has taken the place of x
and y + 16 has taken the place of y t
i.e. instead of x = 0, 2x I = 0, or x = ,
and instead of y = 0, y f 16 = 0, or y = 16,
i.e. the head of the parabola is not (0, 0) but (J, 16) as in
fig. 43. Clearly the graph of 4# 2 4# 15 is identical
 oc
3oc
GRAPHS 163
with the graph of 4# 2 , except that it has been pushed J unit
to the right, and 16 units down. (The scale difference must,
of course, be borne in mind.)
This identification of similar functions is of great im
portance throughout the whole range of algebra. One of
the greatest difficulties of beginners is to see how the form
of a normal function may be obscured by mere intercept
values.
Family of parabolas. Let the boy graph a few related
parabolas like the following: y = x 2 \ y = 2# 2 ; y 3# 2 ; &c.
For 2# 2 , the ordinates of x 2 are doubled; for 3# 2 , tripled;
and so on. Grouping of this kind helps to impress on the
learner's mind the relationship of the curves.
A metal rod bent into the shape of a parabola, with an in
conspicuous crosspiece for maintaining its shape and for mov
ing it about the blackboard, is useful for oral work in class.
Contrast the parabola y ax 2 + bx + c when a is
negative with that when a is positive. With a negative, the
curve is " head up "; e.g. 7 + 3# 4# 2 gives such a parabola.
Fig. 49 shows another. Give the boys a little practice in
drawing parabolas in this position. They should also draw
one or two of the type x = y 2 and x = y 2 , and carefully
note the positions with respect to the axis.
Turning Points. Maximum and Minimum Values
The pupil has learnt that in y = 4# 2 4# 15, the head
of the parabola is (J, 16). He sees that the equation
4# 2 4# 15 has two roots whenever y is greater than
16. For example if y = 9, x = 2 and 3; if y = 7, the
roots are 1 and 2; if y = 15, the roots are and +1.
But if y = 16 the two roots are equal, each being *5. ' If a
line parallel to the x axis is down below y = 16, it does not
cut the curve at all, so that if y is less than 16, x has no
values, or, as is generally said, " the equation has no roots ".
For instance, if we give y the value 17, and work out the
equation 4# 2 4# 15 = 17 in the ordinary way, we
164
CRAFTSMANSHIP IN MATHEMATICS
find that x =
Vi
But these values of x have
no reality because we cannot have the square root of a negative
number. The graph tells the true story. Instead of saying
that the equation has two unreal or " imaginary " roots,
we may more correctly say that, when the value of y is less
than 16, x has no value at all, simply because the y line
" 7
Fig. 48
does not now cut the curve at all. The y line is " out of the
picture ".
As a point moves along the curve from the left downwards,
the ordinate of the point decreases until it reaches the value
16, then a turn upwards is made, and the ordinate begins
to increase as it ascends to the right. The point (+5, 16)
is the turningpoint of the graph, and the value 16 of the
ordinate is called the turning value of the ordinate (or of the
function). That value of the ordinate is its minimum value.
If the graph was one with its head upwards, the turning
point would be at the top and would be a maximum value
(see fig. 49).
GRAPHS
165
Thus the pupil must understand clearly that, in the case
of any parabolic function, the head (vertex) represents a
kind of limiting value of y. Each value of y corresponds to
two different values of x, though the head of the curve seems
to be an exception. Strictly speaking, the head corresponds
to only one value of #, but it is convenient to adopt the con
vention that x has in this case two identical values. Beyond
the head, outside the curve, x can have no values. Some
quadratic equations have two roots, some OPP (two identical),
some none. Do not talk of " imagi.iary ' roots: that is
nonsensical. We shall refer to this point again, in the chapter
on complex numbers (see Chap. XXVII).
The pupil should note how slowly the length of the
ordinate changes near the turningpoint of a parabola. In
fact this characteristic of slow change near a turningpoint
is characteristic of turningpoints in all ordinary graphs.
Let the pupil plot on a fairly large scale y = x 2 for small
values of x.
no
x
^x.
p
I
2 /
?
4
^s
5
X
7
10
/
^
\
to
\
/
s
\
V
f
\
1
\
<(2x 2 !9oc+35)
Fig 49
Show the pupil how the graph tells him at a glance
where the values of y (the function) are positive, say for
y = 19x 2x 2 35. The part of the curve above the x axis
166 CRAFTSMANSHIP IN MATHEMATICS
corresponds to values of x between 2 and 7. But the values of
y (= 19# 2x 2 35) above the x axis are positive. Hence
the expression 19# 2x 2 35 is positive between the values
2 and 7. If any values outside these are tested algebraically,
the expression is seen to be negative. (Fig. 49.)
It may be emphasized again that quadratic equations
should be looked upon as merely one interesting and useful
feature in the general elementary theory of parabolic functions.
Do not forget practical applications of the parabolic function;
e.g. falling bodies in mechanics.
Simultaneous Equations
Practice in solving various types of simultaneous equations
should be given less with the idea of finding the actual roots
of the equations than for the purpose of studying the relative
positions and the intersections of the graphs. We will refer
briefly to two typical examples.
1. Consider the equations:
x z + y* = 97 \ and x* + y* = 20
= 97 \
= 36 /
xy = 36 and xy = 36
# 2 + y 2 = 97 is a circle with its centre at the origin and
radius A/97; and xy 30 is a rectangular hyperbola symmetri
cally placed in the first and third quadrants, with its vertices
at a distance of V2 X 36 from the origin. As V2 X 36 is
less than V97, the circle cuts the hyperbola in four points,
symmetrically placed. In the second case, since V20 is less
than V2 X 36, the circle does not cut the hyperbola, and
there are no roots. (Fig. 50.)
2. Consider the equations:
X = \r
Here we have two parabolas, one with its apex downwards,
touching the axis of x, two units to the right of the origin^
GRAPHS
167
d
,
y
\
^
\^
\
/
/*
\
s^
^
/
X
/
\
/
/
\
o
1
A
\
O
/
\s
S
/
^
\
^~
^
"
A
s
\
3C% o/  97
ocy =36
OV =A/^36; OA  V97
x'H^20
OC'U'  3G
OA  ~/ZO ; OV v'Z x l
Fig. 50
the other symmetrically astride the x axis, with its apex at 1
to the left. The roots are readily obtained approximately by
measurement of the coordinates of the intersections. (Fig. 51.)
jc
4 
3 (
fo.O
(c5 r
2
(20)
2)
 .( X  2)
Fig. 51
168
CRAFTSMANSHIP IN MATHEMATICS
Higher Equations
The pupils should study a few cubics graphically, if only
that they may gain confidence in a method of general applica
tion.
The normal form of the cubic (y = # 3 ) is easily graphed
and remembered.
 re
Fig. 52
Consider the equation S(x 1) (x 2) (x 4) = 0.
Let 8(*  1) (x  2) (x  4)  y.
* =
1
U
2
3
4
5
(*!) =
1
i
1
2
3
4
(*  2) =
2
1
i
2
1
2
3
(*  4) =
4.
3
21
2
j
1
y = 8(*  1) (  2) (*  4) =
64
5
16
96
The curve cuts the x axis at points 1, 2, and 4, which are
therefore the roots of the equation (as, of course, we know
at once from the factors). (Fig. 53.)
GRAPHS
169
(5 16)
^ = 8(xl)(x2)(x4)
Fig. 53
Now consider the equation # 3 7# + 4 = 0.
Since * 7* + 4 = 0; /. * 3 = lx 4.
Let ^ = Y I; 7*  4 = Y 2 ; Y 3  Y t  Y 2  ^ 3  lx + 4.
We will tabulate values for Y a , Y 2 , and Y 3 .
2
2
1
+ 1
+ 2
+ 3
Y, =
x 3 =
27
8
j
+ 1
+ 8
+ 27
Y 2 =
lx  4 =
25
18
11
4
+ 3
+ 10
+ 17
Y 3 
^ 3  7 i 4 =
2
110
+ 10
+ 4
2
2
+ 10
We will now plot Y x (= # 3 ), a normal cubic, and Y 2
(= 7# 4) a straight line, and so solve the equation. The
latter cuts the former in three points, viz. where x = 290,
60, 2'29, which are therefore the three roots. But the
170
CRAFTSMANSHIP IN MATHEMATICS
figure (fig. 54) does not show the graph of y = of  7x + 4,
the original function. To draw this graph, we may either use
the values of Y 3 in the table, or superimpose the above two
graphs, Yj and Y 2 , on each other, remembering that
7
OC
7oc4
Fig. 54
Y 3 = Y t Y 2 , and that therefore we may obtain any ordinate
for Y 3 by taking the difference of the corresponding ordinates
for Y! and Y 2 . For instance, the ordinate at x = 2 is 8
for Y!  a, and 18 for Y 2 = 7x  4, and for Y 3 (= YjYg)
is therefore 8+ 18, or + 10. And so generally. This
y
. 55
time the roots of the equations are given by the intersection
of the curve with the x axis, the values (290, 60, 229)
being, of course, the same as before. (Fig. 55.)
172
CRAFTSMANSHIP IN MATHEMATICS
The Logarithmic Curve
We dealt with the A B C of Logarithms in Chapter XI,
and we now come to the logarithmic curve, the use of which
is, of course, not as a substitute for the tables but as a justifi
cation of the extension of the laws of indices from positive
integers to fractional and negative values. The boy has to
learn, too, that the curve is really a picture of a small set of
tables. He should therefore be taught to plot a curve from
first principles, and to use it as far as he can.
Let him first become familiar with the general form
of the curve. For instance he might plot y = 2 X , 3*, 5*.
Go
50
40
30
20
IO
OC
1
1
1
/
x
'
'
s
 2
l**2
Ex
FiR. 56
3 4
 3
4 8l
T
Ex. 5  125
Show the advantage of changes of scales. Draw two or three
extended logarithmic curves on the blackboard, and spend
a few minutes in oral work, e.g. 2 8 ? 3 6 ? 5 4 ? (approximate
answers are of course, all that can be expected).
The next step is to deal with the evaluation of fractional
indices in y 10*. Let the class graph y 10* up to x = 3,
on a fairly large scale, drawing the graph from the integral
values x = 1, 2, 3. " If the index law holds good, we ought
to be able to obtain by readings from the graph such values
as IO 1 * and IO 2 *. But our graph is necessarily very rough;
GRAPHS
173
we had such a few points with which to plot it. We must
try to construct a better curve.
" Let us use our arithmetic for constructing the curve,
say a curve representing values from 10 to 10 1 . The more
values we find, the more points we shall have for plotting
our curve. How many? Say 7 between 10 and 10 1 , viz.
10s 10*, 10&, 10*, 10s 10s 10&."
Begin with 10* = 10 > =
Then 10*  10'. =
Then 10s =
3162.
1779.
= 1333.
We have 4 more to find, viz.
10s 10, 10s 101.
I0l = (10<) 3 = (1333) 3  2371.
10 == (10<)>  (1333)' = (1333) 2 X (1333) 3
 1779 X 2371 = 4217.
I0l = 10?  (101) 3  (1779) 3 = 5623.
105 = (lO 1 ^) 7  (1333) 7  (1333) 4 x (1333) 3
= 3162 x 2371 = 7497.
If the arithmetic is distributed amongst the class, it is quickly
done; very little explanation is necessary, provided previous
elementary work in powers and roots was understood.
Now the boys can make up their table of values, changing
the vulgar fractions into decimal fractions; then plot their
points, and draw the curve.
X =
125
25
375
5
625
75
5623
875
1
10
y = 10* =
1
1333
1779
2371
3162
4217
7497
The class may now be given a few multiplication and division
sums to work, for the purpose of checking their curve. (Of
course they cannot read to more than 2 places of decimals.)
174
CRAFTSMANSHIP IN MATHEMATICS
1. Multiply 379 x 238.
From the graph, 379 = 10' 6 ' 9 and 238 = 10' 876 .
/. 379 x 238 = 10'" x 10" = 10' 9 = 902 (from the graph).
Now verify by actual multiplication.
*90
JE
3
*
422
i2S~
2L_
178
E%
10
^5"
UC
7&:
_LJA
^Q* 55 .
02
X'
625
ROWERS TO BASE 10.
Fit?. 57
2. Divide 902 by 238.
902 + 238
= 109" .i. XO876 = JQ.955370
Now verify by actual division
Now let the class write into their graph, by interpolation,
the index values of the integral numbers 1 to 10. (Some
GRAPHS
175
teachers make the boys learn off these values to 3 places of
decimals.) The boys' interpolations resulting from their
own measurements will necessarily be very rough and at
this stage a prepared graph of the following kind might be
given them.
X =
301
477
602
699
778
845
903
954
100
y = 10* =
1
2
3
4
5
6
7
8
9
10
/
9
10
954
1
/
ft
= 10
903
/
/
7
10
645
/
/I
G
10
776
./
' \
r
5
10
699
/
/^
4
10
602
/
/^
3
10
477
,
X
/>
^
2
10
301
^
<^
^
^
,
*"
i
50
fA
6C
69
t
77JS
\
an
$
3 *9(
t
,
i
'4
t
f
$
C
>
V
i
i
1
POWERS TO BASE I0 f>
Fig. 58
The term " logarithm " may now be introduced. " It
is just another name for index." Set out a multiplication
176 CRAFTSMANSHIP IN MATHEMATICS
sum in parallel, showing the related methods. Emphasize
the fact that the two things are the same, except in appear
ance.
Multiply 473 by 184.
1.
Let 473 x 184 = x.
x = (473 X 184)
== 10 fl76 X 10 265 (graph)
= IO 940 ;
/. x = 870 (graph).
2.
Let 473 x 184 = x.
log x = log (473 X 184)
= log 473 f log 184
= 075 f 265 (graph)
= 940;
.'. x = 870 (graph).
Now give the boys just one page of 4figure logarithms, make
them work out a few examples in both ways, and see they
understand that the two ways represent exactly the same thing.
It ought now to be possible for the boys to proceed with
logarithms in the usual way, and really to understand what
they are doing.
Graphs and the " Method of Differences "
The nature of a graph may easily be investigated by means
of the method of differences. A series of equidistant ordinates
is drawn, beginning at any point on the graph. The heights
of the ordinates are measured, and a table is made of the
first, second, third, . . . differences. If the graph is a straight
line, the first difference will be constant; if a parabola, the
second difference; if a function of the third degree, the third
difference; and so on. Hence by examining the differences of
the ordinates, we can determine the degree of the function
which corresponds to the graph. This is a useful principle
for the boys to know.
Books to consult:
1. Graph Book, Durell and Siddons.
2. Graphs, Gibson.
ALGEBRAIC MANIPULATION 177
CHAPTER XVIII
Algebraic Manipulation
Common form Factors
During the last 30 years there has been amongst the older
boys of schools a serious falling off in their power of algebraic
manipulation. Nowadays, there is often a sad lack of easy
familiarity with even the simpler transformations in algebra
and trigonometry. Although a great deal of bookwork is
done and mastered, the valuable old transformation exercises
receive too little attention, with the result that there is often
a good deal of uncertainty about everyday working algebraic
procedure.
Readiness in manipulation is the key to algebraic success.
Pupils must acquire facility in the manipulation of common
algebraic expressions.
The factors to be mastered in the first year of algebra are
few, but they are of fundamental importance and must be
taught thoroughly. In the early stages they should be associated
with arithmetic and geometry, if only in order that the
pupils may be convinced of their usefulness.
The early forms are,
ab ac = a(b c),
and a 2  b 2 = (a + b) (a  6);
and the expansions (a + b) z = a 2 + 2ab f b 2
and perhaps (a &) 3  a 3 + 3a*b f 3ab 2 + fc 3 .
Let factors be first looked upon as a device for simplifying
formulae, and for putting these into shape for arithmetical sub
stitution. It is a good plan to begin with obvious geometrical
relations and base upon these an algebraic identity. But do
not talk of " proving " the truth of the geometrical pro
position. The illustrations in Chapter XVI, pp. 1347
typify the kind of thing to be done.
(B291) 13
178
CRAFTSMANSHIP IN MATHEMATICS
The elementary standard forms (a + b) 2 , (a A) 2 ,
a 2 b 2 y being well known, verified by a few numerical
examples, and illustrated geometrically, a first element of
complexity may be introduced into them.
The a and the b may be regarded, respectively, as, say,
a square and a circular box, into each of which we may put
any algebraic expression we please. Thus we may write:
D 2  O 2 = (D + 0)(D  O),
and then fill up, say with p 2 and q 2 respectively, in this way:
.e.
 tf = (p*
Q) (/>
Such a device is very useful, but do not carry such an ex
tension very far at first. Wait a year, and then with harder
examples push the principle home.
The expansions (a b) 3 are probably best postponed
until the second year, though when they are taken up they
should be associated with a geometrical model. A 6in. or 8in.
cubical block, sawn through by cuts parallel to each pair of
parallel faces, makes a suitable model, and may be prepared
in the manual instruction room. Or a cube cut from a bar
of soap may be used, if a very thinbladed knife is available
for cutting the sections. We deal first with the identity
(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 . If each edge of the
cube is cut into two parts a and i, the original edge being
a + b, and a being > 6, the cutup cube evidently consists
of eight portions, viz. a larger cube # 3 , three square slabs
of area a 2 and thickness b, three square prisms of length
a and square section i 2 , and a smaller cube b 3 .
Then the class sees at once that
(a + 6) 8 = a 8 + 3a*b + 3a& 8 + b*.
ALGEBRAIC MANIPULATION
179
But they should discover this identity from the model for
themselves, and not be told.
The same model may be used for the identity
but the manipulation is a little more troublesome. The
whole composite cube must now be called # 3 , and the thick
ness of the other seven parts (slabs, prisms, and small cube)
should be called b. The larger of the two cubes within the
whole composite block is evidently (a A) 3 . When actually
handling the model it is easy to see that this cube (a 6) 3
with the three slabs (3a 2 b) and the little cube (6 3 ) are together
equal to the whole composite block (# 3 ) plus the three prisms
(36 2 ), i.e.
(a  b)^ + 3a*b + 6 3  <z 3 f 3ab\
or (a  b)^  3  Sa*b f Sab*  b*.
The three slabs " overlap ", a fact which tends to perplex
most pupils.
It is really better to cut up two cubes and have two
models, one to be kept in its eight separate pieces, the other
Fig. 59
to be glued up again without the cube (a i) 3 and looking
something like three of the six sides of a cubical box. Unless
the teacher is pretty deft in manipulating such a model, it
had better not be used, or the class will get more amusement
180 CRAFTSMANSHIP IN MATHEMATICS
than instruction from his efforts. It is obvious that since
a AB, the second model (the threesided shell) is less than
the three slabs 3a 2 6 by the three prisms Sab 2 diminished by
the little cube i 3 .
I.e. shell  3a 2 b  (3ab 2  6 3 )
add the removed cube (a 6) 3 to each side:
shell + (a  6) 3  (a  b)* + 3a*b  3ab 2 + b*,
i.e. a 3 = (a  b)* + 3a 2 6  3ab* f 6 3 ,
i.e. (a  b)*  a 3  3a 2 6 + 3a6 2  6 3 (as before).
This on paper looks complicated. With the model in the hand
it may be made clear at once. The case seems complicated
because what we have called a slab a 2 b consists of four pieces
of wood, each of the thickness b, viz. a slab (a b) 2 in area,
two square prisms each (a b) long, and a cube i 3 .
The boys always look upon it as a pretty little puzzle.
Let them build up the cube a 3 themselves, beginning with
the cube (a 6) 3 , and adding and subtracting the other
pieces one by one. The whole difficulty comes about from
calling the edge of the whole cube a as compared with the
previous example when a referred to part of the edge.
A further identity for the boys to discover from their
model is:
The whole cube may be called a 3 and the removable cube
i 3 . Lay out the seven pieces, all of thickness (a 6), on the
table. The united area obviously is:
3ab + (a  6) a
= a 2 + ab + b 2 ;
:. volume = (a b) (a 2 + ab + b*) 9
i.e. a 3  & 3 = (a  b) (a 2 f ab + 6 2 ).
Verify all these identities by a variety of numerical calculations,
and so emphasize the utility of the alternative forms.
It is a curious fact that Form IV boys are prone to forget
ALGEBRAIC MANIPULATION 181
the factors of a 4 f a?b 2 + A 4 . It is a good thing to ask them
occasionally for the factors of (a Q A 6 ). They will give
them readily enough:
= (a 8 + 6 3 ) (a 3  & 3 )
= (a + b) (a 2  ab + b 2 ) (a  b) (a 2 + ab + b 2 ).
Now ask them to multiply the four factors together again,
in pairs:
(a f b) (a b) a 2 b 2 (readily given),
(a 2 ab + b 2 ) (a 2 + ab + b 2 ) (generally forgotten).
If the product is not forthcoming, ask for the factors of
# 4 + a 2 b 2 ~ 4 and give them the hint of adding and sub
tracting a 2 6 2 , thus:
a* + a 2 b 2 + 6 4
= ( 4 f 2a 2 b 2 + 6 4 )  a 2 b*
 (a 2 f b 2 ) 2  W
 (a 2 + ab + b 2 ) (a 2  ab + b 2 ).
Come back to this twice a term, until it is known.
Algebraic Phraseology
Each successive school year will demand its quota of
further manipulative work until in the Upper Fifth, especially
the top Set, the boys become expert. The four or five years'
course of instruction must be organized in such a way that
the difficulties of manipulation are carefully graded. Im
press on the boys that ready manipulation is the key to success
in the greater part of algebra and therefore to the greater
part of trigonometry, conies, and the calculus.
Let your phraseology be accurate, and use it consistently,
exercise after exercise, lesson after lesson, and see that the
boys gradually acquire the use of phraseology of the same
degree of accuracy.
" Jones, what is the first thing to do?" " Rearrange the
terms."
182 CRAFTSMANSHIP IN MATHEMATICS
" How?" " Write down all the plus terms first, and then
all the minus terms."
" Then?" " Put the plus sign . . ."
" No. That is not the way we decided to say things."
" Add up all the plus terms and write down the sum, pre
fixed by a plus sign; then add up all the minus terms and
write down the sum, prefixed by a minus sign."
" Smith: lastly?" " Take the difference between the
two sums, and prefix the sign of the larger."
Remember the slow boys and the amount of practice they
need until the soaking in is complete. Then all is well.
There are certain common algebraic terms which, though
of fundamental importance, are often loosely used. Formal
definitions to be learnt by rote are unnecessary, but con
sistently accurate usage should be adopted from the outset.
Introduce the terms one at a time and make each new one
part of the everyday jargon of each lesson for a few weeks.
We refer to such terms as mononomial, binomial, degree and
dimensions , homogeneity and symmetry, and so forth.
" In algebra, a letter, or a product of two or more letters,
or of letters and numbers, in which there is no addition or
subtraction, is called a term, or a mononomial, e.g. #, # 2 , x 2 y,
" If the same letter occurs more than once in a term
we write the letter down once, and at the top righthand
corner we write a figure to show the number of times it
occurs, e.g. xxx is written # 3 , aaaa is written 4 .
i q
" A term may be integral^ as ab 2 ; or fractional, as .
x
" The degree or the dimension of a term is the sum of the
indices of the named letters; e.g. the term x 2 y 3 is a term of
the fifth degree, or a term of five dimensions.
" A binomial consists of two terms connected by the
sign + or ; a trinomial of three terms; a polynomial of
more than three."
All this is just the stock phraseology of the classroom. But
let it be carefully thought out and consistently used, in order
ALGEBRAIC MANIPULATION 183
that the boy may soon get to know the precise significance
of the new vocabulary.
We have already referred to the term function. Use it
consistently and use it often.
Such a term as the law of commutation is hardly worth
mentioning at all unless it be in Form VI, where algebraic
theory is being minutely discussed. The boys will know
from their arithmetic that the mere order in which terms are
arranged for addition purposes is immaterial. So with
multiplication: the notion of commutation is imbibed with
the multiplication table; 5 sevens gives the same product as
7 fives. Thus, any elaborate formal explanation that
d+c + a + b a { b { c { d, or that b 2 ac is the same
as ab 2 c, is unnecessary. It is, as a rule, enough to point out
the close analogy with arithmetic, though in a firstyear course
of algebra attention must repeatedly be called to the fact
that abc is not in form a faithful copy of 345, and that 345
means 300 + 40 + >. In the main, let early algebraic
processes grow out of corresponding arithmetical processes.
Typical Expressions for Factor Resolution
1. ac + be f ad + bd = (a + b) (c + d).
2. x 2 + (a + b)x + ab = (x + a) (x + 6).
3. acx* + (ad + bc)x + bd = (ax + b) (ex + d).
These depend on a redistribution of terms, and too much
care cannot be paid to the teaching of the principle involved.
We know that
(a + b)(c + d) = a(c + d) + b(c + d)
= ac + ad + be + bd,
and therefore, conversely,
ac + ad + be + bd = a(c + d) + b(c + d)
d).
If then we are given the expression ac + bd + ad + be,
184 CRAFTSMANSHIP IN MATHEMATICS
and we rearrange it so that both the a terms come first, we
have a suitable distribution for finding the factors:
ac + bd + ad f be
= ac + ad f be + bd
= a(c + d) + b(c + d)
= (a + b)(c + d).
Boys are often puzzled about the derivation of the last
line from the last line but one, but their difficulty is cleared
up when it is pointed out to them that if they had to multiply
(a + b) by (c + d), they would begin by writing down
a(c + d) + b(c + d).
Emphasis must be laid on this intermediate step of a
partial redistribution and on how we proceed forwards and
backwards from it.
(a f b \ c) (d + e)
= a(d + e) f b(d + e) f c(d + e)
= ad \ ae j bd f fee + cd + <?,
which /. = a(rf + r) f 6(rf + e) + c(rf f e)
(a + fe + c) (d + e), with which we began.
We append two rather harder examples. It is always a
question of arranging according to the powers of some
selected letter, though which letter only experience can tell.
(i) x* + (a + b + c)x + ab + ac
x 2 f ax f bx + # + fl& + #
Arranging in powers of a, we have
ax + 0fe + ac + # a + &# + #
= ^(A; + b + c) + x(x + b + c)
= (a + x) (x + 6 + c).
(ii) a 2 H 206  2ac  36 2 f 2bc.
We note the letter c in two terms. Try grouping them
together.
Then a 2 + 2ab  3& 2  2ac + 2fo
= (a 2 + 206  36 2 )  2c(a  b)
= (a + 36) (a  b)  2c(a  b)
= (a + 36  2c) (a  6).
ALGEBRAIC MANIPULATION 185
If boys feel a difficulty about accepting the last line as another
form of the last line but one, give them an example of the
reverse kind:
(a + b) (c + d + e)
either = (a + b)c + (a + b)d + (a + b)e
or = (a + b) (c + d) + (a + b)e.
Both redistributions yield exactly the same result.
Illustrate with a numerical example:
47 x 365
either = (47 x 300) + (47 x 60) + (47 x 5)
or = (47 X 360) f (47 X 5),
i.e. we can perform our multiplication in little bits or in
bigger bits, just as we please.
The type x 2 + (a + b)x + ab = (x + a) (x + b) seldom
gives much trouble. Examples:
x 2 + Sx + 15.
x*  8* + 15.
The two rules (1) for signs, (2) for determining the coefficients
of x, should be kept separate. Both admit of very simple
statement.
For the first example we begin by writing (x + ) (# + )>
and for the second example we begin by writing (x ) (x ).
For both examples we ask the question, What two numbers
multiplied together give us 15 and when added together
give us 8? Answer, 5 and 3. Hence the factors (x + 5) (x + 3)
and (x 5) (x 3).
Other examples:
* a + 2x  15
x 2  2x  15.
As the last term is a minus term, the second term of the
two factors will be of opposite signs. Hence we may begin
by writing down for each case (x + ) (x ). " Find two
numbers whose product is 15 and whose difference is 2."
186 CRAFTSMANSHIP IN MATHEMATICS
Answer, 5 and 3. " Give the larger number the sign before
the middle term." Hence we have:
x* + 2x  15 = (x + 5) (x  3)
x*  2x  15 = (x  5) (x + 3).
Of course these are mere rules, to be remembered; but they
should be first worked out from an examination of the different
products, three or four sets being taken for confirmation
purposes.
(x + 3) (x + 5) = x 2 + 8* + 15
(x  3) (x  5) = x 2  Sx + 15
(x + 5) (*  3) = x* + 2*  15
(x  5) (x f 3) = x*  2x  15.
Help the boys to examine the products and to discover:
(1) That if the last term of the trinomial is +, the
signs of both factors are the same, the same as
the middle term.
(2) That if the last term of the trinomial is , the
signs of the two factors are different, the factor
with the larger number taking the sign of the
middle term.
(3) That the last term of the trinomial is always the
algebraic product of the second terms of the
two factors (hence the signs).
(4) That the middle term of the trinomial is always
the algebraic sum of the second terms of the
two factors (hence the signs).
The mere rules must be mastered by all Sets, but experience
shows that the justification of the rules, by an analysis of a
series of products, is beyond lower Sets, though upper Sets
always appreciate them. Do not talk of " proving " the rules.
The type, acx 2 + (ad + bc)x + bd
This common type of expression boys generally find
rather troublesome to factorize. I remember seeing a Fourth
ALGEBRAIC MANIPULATION 187
Form trying to factorize 35# 2 59# 48. There had been
a preliminary discussion on the necessarily long succession
of " trial " factors, and the 33 boys were actually working
out with the patience of 33 Jobs the possible combinations,
the first factors being 35* 1, 7* 1, 5# 1, # 1,
35* 2, 7x 2, 5* 2, x 2, and so on with 3, 4,
6, 8, 12, 16, 24, and 48, 80 possible first factors
in all! Naturally the lesson was not long enough for this
single set of trials to be completed. In any circumstances
the particular example would be very difficult for class
practice. But the " trial " method is unnecessary. All ordinary
cases can be dealt with by a method which is much simpler.
Consider the example (6# 2 + 17* + 12) = (3# + 4)
(2x f 3). Let us multiply the factors together in the ordinary
way.
3* + 4
2x H_3
ftic a ~~+~8jc
4 9* +12
+ 12
We might have multiplied out, thus:
(3* + 4) (2* + 3)
= 2*(3* + 4) + 3(3* + 4)
= 6* 2 + 8x f 9* + 12
= 6* 2 + 17* f 12.
To find the factors, why not reverse this process?
6* 2 + 17* + 12
= 6* a + 8* + 9* + 12
= 2*(3* + 4) + 3(3* + 4)
= (3* + 4) (2* f 3).
Yes, why not? But how could we tell that the ITx in the
first line should be divided into 8# and 9#, instead of, say,
into 3# and 14#, or into 5x and 12#? That is the trouble,
that the only difficulty. How are we to find the two correct
numbers?
188 CRAFTSMANSHIP IN MATHEMATICS
Let us suppose these unknown; call them m and n.
Now m + n = 17 (that we know).
And mn = 72.
[How do we know that? Because 72 is the product of the
6 and 12 which we obtained (in the multiplication sum) by
multiplying 3 by 2 and by multiplying 4 by 3; and from these
same 4 numbers, 3, 2, 4, 3 we obtained the 9 and the 8 also
in the multiplication sum. Thus the 72 is the product of the
6 and 12 in the first and third terms of the trinomial.]
Hence all we have to do is to find two numbers which
when added together come to 17 and which when multiplied
together come to 72. The numbers are easily seen to be
8 and 9, and therefore we now know that the 17* must be
divided into Sx and 9#.
Another example: 14# 2 25* + 6.
Here m + n = 25 and mn = 14 X 6 = 84. By trial,
the two required numbers are 21 and 4.
14* 2  25* + 6
= 14* 2  21*  4* + 6
= 7x(2x  3)  2(2*  3)
= (7*  2) (2x  3).
Another example: 6# 2 llx 10.
This time we have to find two numbers whose product
is 60 and whose sum is 11. The numbers are evidently
15 and +4.
6* 2  11*  10
= G* 2  15* + 4*  10
= 3*(2*  6) + 2(2*  5)
= (3* + 2) (2*  5).
Thus we have this simple rule. Redistribute the terms of
the expression , splitting the coefficient of the middle term into
two parts, m and n, so that m and n is the product of the co
efficients in the first and last terms. Then factorize the re
distributed product in the usual way.
For top Sets the rule can be stated more formally from
ALGEBRAIC MANIPULATION
189
the expression acx 2 + (fld + bc)x + bd, where the relations
stated in the rule are obvious.
At least top Sets should be made to see how, as regards
both coefficients and signs, all the different cases may be
brought under a single rule. Let the general expression be
ax z + bx + c. Then: divide b into two parts m + n so
that win = ac. Now let the class apply the rule to all possible
different cases, say:
x 2 Sx + 15,
x 2 2x  15,
6:c 2 + 19* + 15,
6* 2 + x  15.
Difficult cases where m and n cannot be obtained readily
from mn and m + ;/ at once by mental arithmetic may be
solved quadratically. Examples:
x* + 2x  360.
Write, x 2 + 2x  360 = 0.
/. x 2 + 2x + 1 = 361,
/. x + 1 == 19,
.'. x = 18 or 20,
/. factors = (x 18) (x + 20).
x 2 + 12*  405.
Write, x* + 12x  405 = 0.
/. x* f I2x + 36 = 441,
.'. x + 6  +21,
/. x == 27 or +15,
/. factors = (x  15) (x + 27).
Do not let the pupils look upon these as quadratic equations
but simply as a plan for finding the factors. Quadratic
equations will come a little later. The quadratic principle
may be applied to any case, but more often than not it is
merely a clumsy substitute for the method first mentioned.
For instance, consider 6x 2 llx 10.
6*2 _ 11^ _ 10 = 6(* 2  Y*  )
Solving the quadratic x 2 ~^x J j = 0, we have,
#2 __ i i x __ /i i\2 _ __ /J.i\a
or f .
 11*  10 = 6* 
f)
= (2*  5) (3* + 2).
190 CRAFTSMANSHIP IN MATHEMATICS
Complex derivatives from type forms are a prolific source
of errors with all but the ablest pupils. Much care is necessary
in substituting. Example: factorize 8a 3 (a + 2) 3 .
Type: x*  y* = (x  y) (x* + xy + y 2 ).
Thus x = 2a; y = a + 2.
(2a) 3  (a 4 2) 3
= {2  (a + 2)} {(2a) + 2a(a + 2) + (a + 2)2}
 (a  2) (4a 2 + 2a 2 4 4a + a 2 + 4a + 4)
= (a  2) (7a 2 4 8a 4 4).
Product Distribution Generally
There comes a time, probably towards the end of the
Upper Fourth year or the beginning of the Lower Fifth,
when a boy's accumulated facts concerning products must
be summarized and analysed, and reduced to laws of some
kind. We will run rapidly over the necessary ground.
(a 4 b) (c 4 d). Here we have two factors, each of two
terms. We have to multiply each term of the first factor by
c and then by d and so we have four terms in all, viz.
ac 4 be 4 ad 4 bd.
Compare this with the ordinary arithmetical multiplication.
37 x 24
= (30 + 7) (20 4 4)
= (30 x 20) 4 (30 x 4) 4 (7 X 20) 4 (7 X 4),
and show the close analogy. We have and must have four
products both in the algebra and in the arithmetic.
For similar reasons:
(i) (a 4 b 4 c) (d 4 e) will give 6 products.
(ii) (a 4 b 4 c) (d + e +/) will give 9 products,
(iii) (a 4 b) (c 4 d) (e +/) will give 8 products,
(iv) (a + b + c) (d 4 e +/) (g + h + k) will give 27
products.
ALGEBRAIC MANIPULATION 191
The last will be quite clear if it be observed that each of the
9 products in (ii) has to be multiplied by g, then by h, then
by k. Clearly, then, if the factors consist of p 9 q, and r terms,
the number of products will be p X q X r\ and this will be
quite general. Hence we can tell how many products to expect
in an algebraic multiplication.
But in the above cases, all the terms are different. There
is neither condensation owing to like terms occurring more
than once, nor reduction owing to terms destroying each
other. Either or both of these things may happen.
Consider the product (a f b) (a } b). By the general
rule the distribution will give 4 terms' But only 2 different
letters, a and b, occur in the product, and with these only 3
really distinct products of 2 factors can be formed with them,
viz. a 2 , ab y b 2 . Hence, among the 4 terms, at least 1 must
occur more than once, and, in fact, a X b occurs twice.
The result of the distribution therefore is a 2 + 2ab + b 2 .
Thus we may write:
(a + 6)2 = a 2 + 2ab + b*.
Similarly (a  6) 2 = a 2  2ab + b 2 .
In the case of (a + b )(a b), the term ab occurs twice,
but as the two terms are of opposite signs they destroy each
other. Nevertheless the main rule still holds good: the
product really consists of 2 X 2 or 4 terms.
What are all the possible products of 3 factors that can
be made with the 2 letters a and 6? Evidently
aaa, aab, abb, bbb\
or, a 3 , a 2 6, a& 2 , 6 3 ; 4 in all.
Hence in the distribution of (a + 6) 3 , i.e. of (a + b) (a + b)
(a + b), which by the general rule will give 8 terms, only
4 really distinct terms can appear. What terms recur and
how often?
a 3 and i 3 evidently appear each only once, because to
get 3 a's or 3 6's we must take one from each bracket, and
this can be done in only one way.
192 CRAFTSMANSHIP IN MATHEMATICS
cPb may be obtained:
(i) by taking b from the first bracket, and a from each
of the others;
(ii) by taking b from the second bracket, and a from
each of the others;
(iii) by taking b from the third bracket, and a from each
of the others.
ab 2 : the same holds as for a 2 b.
Thus the result is,
(a + &) 3 = 3 + 3a 2 6 f 3ab* + 6 3 .
Similarly, (a  fc) 3  a 5  3a*b + 36 2  b\
(a + &) 4 = <z 4 
If we remember that the possible binary products of 3
letters, a, b, c, are a 2 , i 2 , c 2 , ab, ac y be (6 in all), then
(a + 6 + c) 2 = a 2 + 6 2 + c* + 2ab + 2ac + 2bc.
The ternary products of 3 letters, a, b, c y are easily enu
merated if we first deal with the letter a, writing down
the terms in which it occurs thrice, then those in which it
occurs twice, then those in which it occurs once; then deal
similarly with 6, for such forms as are not already written
down; then with c. Thus we have (10 in all):
<2 3 , d*by a 2 c, ab 2 , ac 2 , abc,
6 3 , b*c, bc\
(*.
Hence, following the rule, we have:
(a + b + c) 3  (a + b + c) (a + b + c) (a + b + c)
f 3a 2 c + 3ab 2 f
Qabc.
The result may be verified by successive distribution:
(a + b + c) 3 = (a + 6 + c) 2 (a + b + c)
= (a 2 + ft 8 + c 2 + 2ab + %ac f 2bc) (a + b f c)
ALGEBRAIC MANIPULATION 193
Another example: (b + c) (c + a) (0 + b).
Here not all the 10 permissible ternary products can occur,
for a 3 , 6 3 , c 3 are excluded by the nature of the case, a appear
ing in only 2 of the brackets, b in only 2, and c in only 2.
(b + c) (c + a) (a + 6) = fo 2 + b z c + ca 2 + c 2 a f <z& 2 f a 2 6 + 2afo.
But although we do not get the 10 ternary products, we do
get 8 ( 2 x 2x 2) products, according to the general rule.
In the product (b c) (c a) (a b) the term abc
occurs twice but with opposite signs, and there is then a
further reduction:
(b  c) (c  a) (a  b) = be 2  b*c f ca z c*a + ab*  a*b.
S Notation
Upper Sets in the Fifths should be taught to use this
notation. It is easy to understand, though average boys
are a little shy of it. " S " stands for, the sum of all the terms
of the same type as, though its exact meaning in any particular
case depends on the number of variables that are in question.
If 2 variables, a and 6, ^Lab means simply ab.
If 4 variables, #, i, c, d, %ab means ab + ac + ad
+bc + bd + cd.
If 2 variables, a, i, 2a 2 6 means a 2 b f ab 2 .
If 3 variables, a, b, c, *Ld l b means a*b + ab 2 + a * c
+ ac 2 + b 2 c + be 2 .
The context usually makes clear how many variables are to
be understood.
" Choose any one of the terms and place S before it."
The use of the sign certainly saves labour: thus
(a + &) 3 = Sa 3
(a + b + c) 9 = Sa 3 + 3Sa 2 6 + Gabc.
(b + c) (c 4 a) (a + b) = Z.a*b f 2o6c.
(E291) 14
194 CRAFTSMANSHIP IN MATHEMATICS
More Complex Forms
Quick boys soon pick up the method of manipulating
more complex forms based on those already familiar to
them, but the slower boys require much practice and should
not be worried by forms so complicated as to be puzzling.
The slower boys should always first be given forms with +
signs only. The added difficulty of negative signs should
come a little later.
(i) Type forms, with the addition of coefficients; e.g.
(a) (3a + 2ft) 3  (3) 3 + 3(3a) 2 (2b) + 3(30) (2ft) 2 f (2ft) 3 .
(P) (a + 2ft + 3c) 2 = &c.
(ii) Type forms with a mononomial replaced by a
binomial; e.g.
Replacing b throughout by (b f c) y we have
(a + b + c) 3 = a 3 + 3a*(b + c)
= &c.
(iii) Association of parts of factors of more than 3
terms; e.g.
(a) (a + b f c  d) (a  b + c + <*)
 {(a + *) + (6  <0) {(a + c )(b d)}
 (a f c?  (b d)\ &c.
(P) ( + 6 + c) (6 + c  a) (c + a  b) (a + b  c)
= {(b + c) + a} {(b + c) a}{a~(b c)}{a + (b  c)}
 {(b + c) 2  a 2 } (a 2  (6  c) 2 }
= (6 2 + 2bc + c 2  a 2 ) ( 2  fe 2 f 2bc  c 2 )
= {26c + (6 2 + c 2  a 2 )} {2fo  (6 2 + c*  a 2 )}
= (26c) 2  (6 2 f c 2  a 2 ) 2
 a 4  fc 4  c*.
The type forms are few and are easily remembered, and
all boys should have them at their fingerends. The possible
applications and developments are, of course, very diverse,
ALGEBRAIC MANIPULATION
195
but do not perplex boys with expressions that are beyond
their manipulative skill at any particular stage.
Detached Coefficients. First Notions of Manipulation
Here is a useful general theorem, easy for upper Sets to
remember. If all the terms of all the factors of a product be
simple letters unaccompanied by numerical coefficients and
all +, the sum of the coefficients in the distributed value
of the product will be / X m X n . . . , where /, m y n, are
the numbers of the terms of the respective factors.
Thus in the evaluated products of the following, we have:
Coefficients.
Sum of
Coefficients.
(a + b)*
1 + 2+1
4 = 2 2
(a + b) 3
1+3+3+1
8 = 2 3
(a + &)
1 + 4 + 6 + 4 + 1
16 = 2*
(a + b) 5
1 + 5+10+10 + 5+1
32 = 2 5
(b + c) (c + a) (a + b)
1 + 1 + 1 + H 1 + 1 + 2
8 = 2 s
The theorem is useful in connexion with expansions.
Simple Expansions and First Generalizations therefrom.
Let an upper Set in a Fourth Form obtain the following
results by continued multiplication, the second being obtained
by multiplying the first by (x + #), the third by multiplying
the second by (x + a) y and so on.
(x + a) 2 = x 2 + 2xa + a 2 .
(x + a) 3 = x 3 + 3* 2 a + 3
(x + a)* = x 4 + 4x 3 a +
(x + a) 6  x 5
(x + a) 6  x 6
a 3 .
+ 4xa 3 +
15* 2 a 4 + Gxa* + a 6 .
Now help the boys to generalize, and to establish the usual
rules. Afterwards, they may work out a few higher expansions
and see that these follow the same rules.
1. The power to which we have carried (x + a) gives
196 CRAFTSMANSHIP IN MATHEMATICS
the index of the highest terms of the expansion and is there
fore the degree of the function.
2. The function has one term more than that index.
Thus the expansion (x + #) 4 has 5 terms.
3. The powers of x run in descending order from the
first term to the last term but one; the powers of a run in
ascending order from the second term to the last. (There is
no objection to writing the first term x n a and the last x Q a n >
if the class understand that x a = 1. Then both x and a
appear in every term.)
4. The dimensions of all the terms are the same and are
always equal to the power to which (x + a) is carried.
5. The coefficients follow a regular law. We may detach
them from their terms (detached coefficients may often be
usefully considered alone), and place them in order, thus:
(* + 0) 1
(* + *) 2
(x + a) 3
(* + aY
(x + a) 5
(x + a)*
I 1
1 2
1
1
6
1 3
1 Y~
1 5
1 6
3 1
6 4 1
10 10 5
15 20~~~~15
The sum of any 2 successive coefficients in any line gives
the coefficient standing in the next line immediately below
the second of these. Thus, in the third line the 6 is the sum
of 3 and 3 in the second line; in the last line, the 15 is the
sum of 10 and 5 in the fifth line. Show that this is the simple
result of continued multiplication. For instance, if we multiply
(x + fl) 3 by (x f a) we have:
1 +3+3+1
1 + 1
1+3+3+1
1+3+3+1
1+4+6+4+1
The second partial product is arranged one place to the right
under the first partial product. Thus any coefficient for any
ALGEBRAIC MANIPULATION 197
expansion may be found by taking the coefficient of the
corresponding term in the previous expansion and adding
to it its predecessor. Let the boys continue the table: they
like the work. They soon see that when they have written
the expansion of, say, (x f a) 10 , they can immediately write
down that for (x + 0) 11 ; it is merely a question of carrying
on the game already begun.
" There is something still more interesting to learn about
the coefficients. Consider the expansion of (x + of. The
coefficient of the second term is 5; we may write it {. The
coefficient of the next term is 10, which we may write ^.
That of the next term is again 10, which we may write
1x2x3* ^ nc * so g enera ^y Examine the other expansions
and see if a similar rule is followed; for instance, (x + a) Q
6 * , 6.5 A , 6.5.4 , . 6.5.4.3 9 .
5 * 4
/ , N
(x + a) 6 =
  __
1.4 L.^i.o l.Z.o.4
6.5.4.3.2 . 6.5.4.3.2.1 6
With one or two leading questions, the boys will see that the
coefficient of a 6 is the same as that for X Q , that for xa 5 the
same as for r r> a, that the coefficient ^ is the same as
c ~^, and so generally. Let them formulate the obvious rule
for themselves.
Let them write down the first few terms of such an
expansion as (x + fl) 20 
First they write the terms without coefficients:
x 20 + x l9 a + x l *a* + &c.
Then they work out their coefficients and insert them:
. ,20 . 20.19 , Q
The object of all this is not to teach the Binomial Theorem:
that will come later. It is to impress boys with the wonderful
simplicity and regularity that underlies algebraic mani
198 CRAFTSMANSHIP IN MATHEMATICS
pulation. Never mind the generalized form (x + a) n , yet.
Never mind the general term. Never mind nCr. When
these things are taken up later, the way will have been paved,
and the work will give little trouble.
The Remainder Theorem
This theorem must be known in order that the Factor
Theorem may be clearly understood. It may be approached in
this way. We know that (x 5) is a factor of (x 2 \x 30),
and in order to find out the other factor we may conveniently
set out the process of division, exactly as in arithmetic.
x  5)* 2 + x 30(# + 6
x 2  5*
0* 30
6*  30
Of course there is no " remainder " (R), but if (x 5) was
not a factor there would be a R. Divide (3# 2 2x + 4)
by (x  5).
x  5)3* 2  2* f 4(3* f 13
??L_Z. I5x
13*+ 4
13*  65
69 = R.
The remainder is 69, and by analogy with arithmetic we know
that
Dividend = (Quotient X Divisor) f R.
Suppose we had to divide (x 2 + px + q) by (x a).
x a)x* px + q(x + (a p)
x 2 ax
x(a p) + q
x(a  p)  a(a  p)
a(a p) f q = R.
Note that a(a p) + q really is the remainder, for it does
ALGEBRAIC MANIPULATION 199
not involve x, and we cannot proceed with the division any
farther.
Now let us set out in this way the previous example,
treating the figures as if they were letters, and not actually
multiplying and subtracting as we did before.
x  5)3x 2  2x + 4(3* + (3.5  2)
x(3.5  2) + 4
x(3.5  2)  5(3.5  2)
5(3.5  2) + 4 = R.
As might be expected, the R is the same, viz. 69.
Now compare the Remainders and the Dividends in both
the last examples.
Dividend = x 2 px + q. I Dividend = 3# a 2x + 4.
Remainder = a 2 pa + q. \ Remainder = 3.5 2 2.5 + 4.
Clearly, then, if the remainder was the only thing we wanted,
we could have written it down at once, for it is exactly of the
same form as the dividend. We merely have to substitute
for the x in the dividend the second term of the divisor
(a in the first case, 5 in the second), treating these, however,
as if they were positive.
Give the pupils several examples, and convince them of
the truth of the rule.
The Remainder " Theorem ", as it is called, provides us
with a simple means of calculating the remainder of a particular
kind of division sum in algebra, without actually performing
the division.
The particular kind of division sum is that in which the
divisor and the dividend are functions of the same letter
(say #), and in which the divisor is a linear expression such
as (x 5) with unity as the coefficient of x.
Example: if we divide
(* 3  7* 2 + 5*  1) by (x  9), the R is
(9 3  7.9 2 + 5.9  1) = 206.
We have merely substituted 9 for x in the dividend.
200 CRAFTSMANSHIP IN MATHEMATICS
Hence:
(1) The Theorem. When a function of x is divided
by (x a), the R is obtained by substituting
a for x in the function.
(2) The why of it. We know that,
Dividend = (Quotient X Divisor) f R.
If we make a equal to x, the divisor (x a) = 0.
/. Dividend = (Quotient X 0) f R,
R,
i.e. by substituting a for x in the Dividend, we have the R.
The Factor Theorem
What is the remainder when we divide (# 2 7x f 10)
by (x  5)?
Substituting 5 for x in x 2 Ix  10,
we have 5 2 7.5 \ 10
 0.
Since R = 0, (x 5) divides exactly into (x 2 7x + 10)
and is therefore a factor of it.
Thus we have a method of finding out whether an ex
pression of the type (x a) is a factor of a given expression
Example: Is (3* 3  2* 2  7*  2) divisible by (x  2) (x + 1)?
Writing 2 for x in the first expression, we have
24 814  2 = 0. Hence (x 2) is a factor,
Again, writing 1 for x we have
32 + 720. Hence (x + 1) is a factor.
Homogeneous Expressions
A homogeneous expression is one in which all the terms
have the same dimensions; e.g.
x* + xy + y*> or a 3 + 6 3 + c* + Sabc.
ALGEBRAIC MANIPULATION 201
It may sometimes be necessary to state what letters are re
garded as giving dimensions; e.g. x 3 + ax 2 y + 2xy 2 + 3y 3
is homogeneous in x and y but is not homogeneous if a is
regarded as having dimensions.
Obviously the product of two homogeneous expressions
is itself homogeneous.
The only homogeneous integral functions of x and y
of the first and second degrees are,
Ax 4 By,
Ax 2 4 "Rxy + Qy 2 .
For 3 variables the corresponding functions are,
Ax  I*y 1 C#,
Ax 2 4 By 2 4 C* 2 4 D;vx? + Esx + Fxy.
The class may usefully write down functions of the third
degree. Upper Form boys should be thoroughly familiar
with all such general expressions.
Symmetry
A function which is unaltered by the interchange of any
two of its variables is said to be symmetrical with regard
to these variables', e.g. x 2 xy 4 y 2 is symmetrical with
regard to x and y] (y + #) (z + x) (x \ y) is symmetrical
with regard to x, y, and z. But x 2 y + y 2 z 4 ~ 2 # is not a
symmetrical function of x, y, and #, for the 3 interchanges
x with y, y with z, z with x, give, respectively,
y z x + ff 2 * + # 2 }>,
x 2 z + # 2 ;y 4 :V 2 #
z*y 4 y 2 * 4 A,
and although all these are equal to each other, none of them
is equal to the original expression.
But the term " symmetry " is not used in quite the
same sense by all writers in algebra. " Cyclic symmetry "
expresses a much clearer connotation.
202 CRAFTSMANSHIP IN MATHEMATICS
Cyclic Expressions
An expression is said to be " cyclic " with regard to the
letters a y b, c, d, . . . k> arranged in this order, when it is
unaltered by changing a into b, b into c, . . . k into a. Thus
the expression a 2 b + b 2 c + c 2 d + d 2 a is cyclic with regard
to the letters a, i, , d, arranged in this order, for by inter
changing a into b, b into c, c into d, d into a, we get
b 2 c f c 2 d + d 2 a + a 2 b, the same as be
fore. Note that the first term is changed
to the second, the second to the third,
and so on. It is merely a question of
beginning at a different point on the
circle, but always going round in the
same direction.
Fig. 60 If in a cyclic expression a term of
some particular type occurs, the terms
which can be derived from this by cyclic interchange must
also occur, and the coefficients of these terms must be equal.
Thus, if x y y, and % are the variables, and the term x 2 y
occurs, then all the terms x 2 y y x 2 z y y 2 z, y 2 x y % 2 x, z*y must
occur. The cycle is easily seen if the six terms are thus
collected up:
x*(y + *) + y*(* + x) + z\x + y).
Expressions which are unaltered by a cyclical change of the
letters involved in them are called cyclically symmetrical.
Thus (b c) (c a) (a b) is cyclically symmetrical, since
it is unaltered by changing a into i, b into c, and c into a y
that is " by starting at a different point in the circle ".
Legitimate Arguments from Cyclical Symmetry
Find the factors of
a\b  c) + b*(c a) + c\a  b). . . . (i)
Here is the solution from one of the best textbooks we have.
ALGEBRAIC MANIPULATION 203
" If we put b = c in the expression, the result is zero,
and it therefore follows from the Remainder theorem that
(b c) is a factor.
" In a similar manner we can prove that (c a) and
(a b) are factors.
" Now the given expression is of the third degree; it
can therefore have only 3 factors.
" Hence the expression is equal to
N(6  c) (c  a) (a  6), (ii)
where N is some number which is always the same for all
values of a, ft, c.
" We can find N by giving values to a, ft, c, Thus, let
a = 0, ft = 1, <: = 2; then (i) = 2, and (ii) = +2N.
Hence N = I.
" Therefore the factors are (b c) (c a) (a ft)."
This argument is open to criticism. It is wholly un
necessary to say, " in a similar manner we can prove that
(c a) and (a ft) are factors". Once we know that (ft c)
is a factor, it follows at once that (c a) and (a ft) are
factors. What applies to (ft c) must apply to (c a) and
(a ft). This is the very essence of cyclical symmetry.
Nay, it is the very essence of all algebraic manipulation.
That (c a) and (a ft) are also factors requires no argu
ment of any sort or kind, except, " it follows from cyclical
symmetry "; and no further argument should be tolerated.
Another example. Find the factors of a 3 (b c) + b\c a)
+ c 3 (a ft). As in the last example (ft c), (c #), (a ft)
are all factors. Now the given expression is of the fourth
dimension; hence there must be a fourth factor, and that of
the first dimension. Since this factor must be symmetrical
with respect to #, ft, , it is necessarily (a + ft + c). Thus
the required factors are
N(6 c)(ca)(ab)(a + b + c),
N being found in the usual way. Any sort of more elaborate
process or argument should be sharply criticized.
204 CRAFTSMANSHIP IN MATHEMATICS
Another example. Find the product of (a + b + c)
(a* __ #2 _j_ c z _ fa _ ca __ a b^
Each of the two factors is symmetrical in a, 6, c, and
therefore the product will be symmetrical in a, 6, .
Obviously the term a 3 occurs with the coefficient unity;
hence the same must be true of 6 3 and c 3 .
Obviously, too, the term Ire has the coefficient 0; hence
by symmetry the five other terms Ire, c 2 a y ca 2 , ab 2 y a 2 b belong
ing to the same group must have the coefficient 0.
Lastly, the term abc is obtained (i) by taking a from
the first bracket and be from the second; hence it is also
obtained (ii) by taking 6, and (iii) by taking c, from the first
bracket. Thus the term abc must have the coefficient 3.
Hence the product
= a 3 + fe 3 + c 3  3abc.
Boys should gain complete confidence in arguments from
symmetry. In at least the A Sets of the Fifth Form, cumbrous
processes should be prohibited whenever arguments from
symmetry are possible.
Identities to be Learnt
The following identities should be at the finger ends of
all Fifth Form boys.
1. (bc) + (ca) + (ab) = 0.
2. a(b  c) f b(c  a) f c(a  b) = 0.
3. (a + b  c) 2 = a 2 f b 2 + c 2 + 2a6 + 2bc f 2ca.
4. (a + b f c) 3  a 3 4 6 3 f c 3 + 3b 2 c + 3bc 2 + 3c 2 a + 3oi a
f 3a 2 Z> f M 2 + Gabc.
5. (a + b + c) (a 2 + b 2 + c 2  be ca ab) = a 3 f ft 3 f &
Zabc.
6. (6  c) (c  a) (a  b) =  2 (6  r)  2 (c  a)  c 2 (a b)
= fo(6 c) ra(c a) ab(a b).
1. (b + c) (c + a) ( + 6)  a(6 + c) + b 2 (c + a) + c 2 (a + 6)
ALGEBRAIC MANIPULATION 205
and perhaps,
8. (a + b + c) (a 2 + 6 2 + c 2 )
= bc(b + c) 4 ca(c 4 a) + ab(a + b) + a 3 + b 3 4 c 3 .
9. (a + b 4 c) (be 4 ca 4 aft)
= 2 (6 4 c) 4 ft 2 (c + a) 4 2 (a + ft) 4 3fo.
10. (a 4 ft 4 c) (ft 4 c  a) (c 4 a  ft) (a 4 6  c)
= 2ft 2 c 2 4 2c 2 a 2 4 2a 2 ft 2  a 4  ft 4  c 4 .
Books to consult:
1. Textbook of Algebra (2 vols.), Chrystal (still the leading work
in the subject).
2. A New Algebra, Barnard and Child.
CHAPTER XIX
Algebraic Equations
Equations of Different Degrees
" Either . . . or?"
More than once I have heard a small boy round on his
teacher for this kind of argument:
Solve the equation x 2 7x + 12 = 0. " Factorizing, we
have (x 4) (x 3) = 0. Hence either (x 4) or (x 3)
must be zero, i.e. x must be either 4 or 3, and therefore
both 4 and 3 must be roots of the equation."
Says the boy: " You said either (x 4) or (x 3) must
be zero; how then can it follow that x is both 4 and 3?"
The criticism is just, for the reasoning is faulty.
A formal approach to equations may be successfully
made by such general arguments as follows.
It is advisable in the first place to distinguish between
an equation and an identity, and consistently to use the same
form of words when referring to them. For instance: "When
two expressions are equal for all the values of the quantities
206 CRAFTSMANSHIP IN MATHEMATICS
involved, the statement of their equality is called an identity"
e.g. that m (n p) = m n { p is true for all values of
the letters m, n y and p.
" But when two expressions are equal for only particular
values of the quantities involved, the statement of their
equality is called an equation." Thus # + 7 = 10 is an
equation; it is true only where x 3.
If in an equation we bring all the terms from the right
hand side to the lefthand side, and equate the whole to 0,
e 8 x + 7  10 = 0,
then by giving x its own particular value, the expression
" vanishes ", e.g.
3 + 7 _ 10 = 0,
i.e. 3 + 7 10 is seen really to be 0, and has therefore
" vanished ".
The value of the unknown quantity that makes the two
sides of an equation equal is said to satisfy the equation.
The process of finding that value, the root, is called solving
the equation.
Consider the equation 3(# 2) = 2(x 1).
For what value of x is 3(x 2) equal to (2x 1)?
Try a few values, say the numbers 1 to 10. The only one
of the ten that makes the expressions equal is 4, i.e.
3(4  2) = 2(4  1),
and so we say that 4 is the root of the equation.
If we simplify the original equation, we have
3*  6 = 2x  2,
/. 3*  2x = 6 2,
.". x 4, as expected.
We may write #=4 as x 4 =^ 0, and when in the ex
pression x 4 we write 4 for x, the expression vanishes,
for 4  4 = 0.
Again, for what value does x 2  x = 6?
Try a few numbers as before. We find that in this case
ALGEBRAIC EQUATIONS 207
there are two values which satisfy the equation, viz. 3 and
2, and that there are no others. Substituting, we have
3 2  3  6,
and (2) 2  (2) = 6.
If we write the equation in the form x 2 x 6 = 0, the
expression on the lefthand side vanishes when we write
either x = 3 or x = 2. Thus
3 2  3  6 = 0,
and (2) 2  (2)  6  0,
and it does not vanish for any other value.
With equations of the second degree, we may always
find two values of x that will satisfy the equation.
Since x 2 x 6 = 0,
and since x* x 6 = (x 3) (x f 2),
.'. (x  3) (* + 2) = 0.
Now a product cannot be equal to zero unless one of the
factors is equal to zero, and hence (x 3) (x +2) can be
equal to zero only (1) when x 3 0, and (2) when x + 2
= 0, and never otherwise. Thus when we have (x 3)
(x + 2) = 0, we may equate the two factors equal to
separately, solve the two simple equations, and obtain the
two roots:
Since x 3 = 0, .*. x = 3,
and since x f 2 = 0, .*. x = 2; as before.
Thus we have a method of solving a quadratic equation.
Bring all the terms to the lefthand side and equate to 0;
break up the expression into factors, and equate each of these
to 0; solve the two resulting simple equations.
Note that a quadratic equation has two roots.
Here is a quotation from a wellknown textbook: " If
the product of two quantities is nothing, one of the quantities
is nothing." One objection to this phraseology is that " noth
ing " is not a mathematical term.
Suppose we have an equation of the third degree, a
208 CRAFTSMANSHIP IN MATHEMATICS
cubic equation as it is called, say, x* 6# 2 llx = 6.
Bring all the terms to the lefthand side and equate to 0.
By a series of trials we may discover that there are 3 and
only 3 values of x which will make the expression on the left
hand side vanish, viz. 1, 2, and 3 Thus
# 3  6* 2 I 11* 0 =
I 3 6(l) 2 + 11 6 =
23 _ 6(2) 2 + 2260
33 _ 6 (3)2 43360.
Hence the roots of the equation are 1 , 2, 3. But the trials
would have been tedious. Let us factorize as before.
Since x*  6* 2 + 11*  6 == 0,
(x !)(* 2) (*  3) = 0.
A product of factors can be equal to only when one
of its terms is equal to 0. Obviously, in the equation, this
may happen in three different ways, when (x 1) = 0, when
(x 2) = 0, when (x 3)  0, and in no other way. If
then we solve these three simple equations, we get x = 1
or 2 or 3, as before. A cubic equation has three roots.
Suppose we have an equation of the fourth degree, a
" biquadratic " equation as it is called. It is quite easy to
solve if we can factorize the expression made by bringing
all the terms to the lefthand side. Usually this is a difficult
job, but here is an easy one.
x* + Qx* f 38* = 8jc 3 + 40,
/. #*  8* 3 + 9* a + 38*  40  0,
/. (x !)(* + 2) (x  4) (x  5)  0.
This product can be zero only if one of its factors is zero.
This can happen in 4 ways, and only in 4, viz. when x 1 == 0,
#j2 = 0,# 4 = 0, x 5 = 0. Thus the orginal equation
is equal to these 4 separate simple equations, and the roots are
1, 2, 4, 5.
Thus a biquadratic equation has 4 roots. And so we
might go on.
The general rule for solving an equation of a degree
ALGEBRAIC EQUATIONS 209
beyond the first is, then, to bring all the terms to the left
hand side, to reduce the resulting expression to a series of
linear factors, to equate each of these to 0, and then to solve
them as simple equations.
It is therefore clear that the roots of an equation of any
degree may be written down at once, provided we can resolve
into linear factors the expression which results from bringing
all the terms of the equation to the lefthand side. Generally
speaking, the trouble is to find the factors, and it is often
necessary to resort to indirect methods.
The Need for Verifying Roots
When solving equations, we frequently adopt the device
of multiplying or dividing both sides by some quantity,
and sometimes we square, or take the square root of, each
of the two sides. Is this always allowable?
Consider an equation of the simplest form, one having only
one solution, say x 3 = 2. Since x 3 2, x 5 =0,
and .. x = 5. Let us multiply both sides of the original
equation by, say, (x 6). Thus
(x  3) ( X  6) = 2(*  6),
/. x 2 9x f 18 = 2.v 12,
/. x 2  llx H 30  0,
/. (x  5) (x  6) = 0,
.". the values of x are 5 and 6.
Thus by introducing the factor (x 6) we have trans
formed the equation into another completely different. The
new root 6 does not satisfy the original equation. Evidently
when we have solved an equation we must see if the roots
really satisfy the equation.
Another example: consider the very simple equation
x = 3. Square both sides,
* 2 = 9,
/. *2 _ 9 = 0,
/. (x + 3) (x  3) = 0.
Hence there are 2 roots, +3 and 3, as compared with
(B291) 15
210 CRAFTSMANSHIP IN MATHEMATICS
only one root (+3) in the original equation. Thus the
squaring has introduced an extraneous root.
Another example:
3* V* 2 ^~2~4 = 16.
/. 3*  16 = V^24
:. 9* a  96* + 256 = x 2  24
/. 8* 2  96* f 280 = 0,
.'. x 2  I2x + 35 = 0,
... (x  7) (*  6) = 0,
/. x = 7 or 6.
But on examination we find that only 7 satisfies the original
equation; 5 does not. Hence there is only one root. We
seem to have solved the equation in the usuial way: have we
done anything wrong? Let us see if by working our way back
wards we can discover any sort of mistake.
(x  5) (x  7) = 0,
/. x 2 I2x + 35 = 0.
Multiplying by 8, 8* 2 96* f 280 = 0.
Adding x 2 24 to each side, 9* 2 96* + 256 * 2  24.
Extracting the square root of each side, 3* 16 = + v/ * 2 24.
.'. 3* T v ^l4 = 16.
The steps are exactly the same until we come to the last but
one; then we had to prefix the double sign. Hence at the
second step in our forward process, we really introduced a
new and extraneous root, since the square of + V# 2 24
is also the square of V# 2 24! Thus from that step on
wards, the equations ceased to represent the original equation.
When we multiply or divide by ordinary arithmetical
numbers, no difficulty will arise. When we multiply or
divide by an algebraic expression, we sometimes run a risk.
What is wrong with the following, for instance?
Suppose * = y.
Then x 2 = ocy,
.'. x 2 y 2 xy y* 9
.' (x + y) (x  y) = *(*  y),
:. x + y = *,
/. x f x = x,
:. 2=1, which is absurd.
ALGEBRAIC EQUATIONS 211
We have divided both sides in the fourth line by (x y),
i.e. by (x x), i.e. 0. This is quite illegitimate, and it in
evitably leads to an absurdity. Can you see now why our
first example went wrong? We had, really, (x 5) = 0;
and then (x 5) (x  6) = Q(x 6), though we did not
show it this way.
Another example:
2 x , 1 A
f 2 4  =0.

x 2 1 x 1
Multiply by * 2  1, the L.C.M.,
x* ~ 3x f 2(x*  1) + * h 1 = 0,
/. 3* 8  2x  1  0,
/. (3* 4 1) (x  1)  0,
.*. x = J and 1.
But by testing we find that 1 is not a value of the original
equation and must therefore be rejected. Multiplying by
(x 2 i) led to this trouble. Here is a more correct way of
solving:
 3*
2 = 0,
^ + 2 = 0,
j o rv
.". x = 5, the only root.
The former method is quite acceptable, provided the roots
found are checked, and that one is rejected if found un
acceptable.
Strictly speaking, either ... or are " disjunctive "
they therefore suggest alternatives. But sometimes they
212 CRAFTSMANSHIP IN MATHEMATICS
are equivalent to both . . . and] or, alike . . . and\ and it
is this exceptional use which more correctly represents the
algebraic argument. But the use of either ... or in con
nexion with equations is best avoided. When a boy says
" either 6 or 5 " he naturally thinks that if one is accepted
the other is necessarily rejected.
The Theory of Quadratics
The work on equations should be closely associated with
the work on graphs. The graph helps to elucidate all sorts
of difficulties. See, for instance, fig. 46, p. 161, in connexion
with the " either ... or " argument.
The elementary theory of quadratics, as far as it is
necessary for a Fifth Form, seldom gives trouble. The more
elementary facts should be known thoroughly and should
be consistently used for checking and other purposes. But
do not forget that the quadratic function is of far greater
importance than the quadratic equation (see the chapter on
Gra p s) 
.  , ,
The formula  ==  may be used as other
2a
formulae are used, but it should not be used as the stock
method of solving quadratics; boys are apt to forget its
significance if used in that way. They should clearly realize
that the formula represents the roots of the equation ax 2 + bx
+ c 0, and that these roots are real and different, real
and equal, or unreal and different, according as the dis
criminant b 2 4.ac is +, 0, or .
The pupils should frequently make use of the further
facts that if x l and x 2 arc the roots of the equation ax 2 + bx
+ c = 0, then x 1 + x 2 = , and x^ = . The method
a a
r i  u r * Vb 2 4ac j ,
or evaluating these from   should be re
membered. a
ALGEBRAIC EQUATIONS 213
Equations Solved like Quadratics
Group the common types together. They are usually
easy, though attention must be paid to all the roots involved;
the boys are apt to overlook some of them. We append
examples of the main type, and add the sort of hint that
ought to suffice to enable the boys to set to work.
1. x l
Write * 2 = y, and solve for y\ then from the 2 values of y
obtain the 4 for x.
2. (x 2 + 2) 2  29(* 2 J 2)  198  0.
Write # 2 + 2 = y and solve for y.
3. 2x z  4x f 3 Vx 2 ~2x \ 6  15 = 0.
/. 2(x*  2* f 6) + 3V* 2 ^"2* +7>  27 = 0.
Write Vx 2 2x 4 6 = y and solve for y.
4. (x l)(x 3) (x  5) (x  7) = 9,
.'. (x !)(* 7) (x  3) (x  5)  9  0,
/. (x*  8* + 7) (^c 2  8^ + 15)  9 = 0.
Write x 2 8x = y, and solve for y.
x f 4 A: 4 9 + ^: 9 A:
5.
 4 i + 4 9 "i 9 +
16* 36*
16 81 x 2
4 9
6. X*+   11 0,
/ **2
(obviously * is a root),
(obviously 2 more roots).
/. **  II* 2 f 18 = 0.
7. x z + 1 = 0. (Factorize.)
8. x 6 7x* 8 =r 0. (Factorize.)
9. 7x*  I3x z + Zx + 3 = 0. (Factorize: (*  1) evidently
a factor.)
Boys soon see through all these types and solve examples
fairly readily.
214 CRAFTSMANSHIP IN MATHEMATICS
Simultaneous Equations
Do not spend much time over these, unless you are un
lucky enough to have to prepare for an unintelligent exam
ination in which farfetched examples are given. Let a
result of each main type be graphed (see the Chapter on
Graphs) and all the roots be pictorially explained.
Teach the pupils to pair off the roots correctly.
We append an example of the commoner types.
1. x \ y = 1
XV 14 Show that once we know the value of x \ y
and x y, we may obtain the separate
2. x 2 f y 2  1^ values of x and y by mere addition and
xy 6 subtraction; and that x \ y can always be
obtained from x 2 f 2xy ~ y 2 , and x y
3. X  y = 2 from x * _ 2xy + yZf
4. x 3 ! y 1 = 152
X \ y  S By division we obtain quotients which
enable us to proceed as in examples
5. *3j,3 = 98 Ito3 .
x 2 f xy f y 2 = 49
Ity division, we obtain x 2 ~ xy + y a ,
6. ^ 4 + ^j; 2 \y*  133 which, subtracted from x 2 f ^
Jt: 2 4 ^ry + jy 2 ^ 19 I y 2 , gives us ;ry. Then as in
examples 1 to 5.
7. 3x* + 4xy + 5); 2 ^=31 From the second express y in terms
X ~\ 2y 5 of x, and substitute in the first.
Expressions are homogeneous. Con
vert into fractions, simplify, and
8. X 2 + 3xy y 2 = 9 factorize. Thus we have:
2x 2  2xy f 3y 2 = 7 ll* a  39^ + 34>' 2 = 0,
/. (11* lly) (x  2y) = 0,
.'. x = ^{y and 2y\ &c.
All these types are easily taught and remembered. It is
waste of time for boys to be given the farfetched and
exceptional types worked out (often elegantly it is true)
in the textbooks. School life is not long enough.
ALGEBRAIC EQUATIONS 215
Problems producing Equations
These have been given a place greatly beyond their value,
and important mathematical principles are often treated
rather superficially in order that more time may be devoted
to " problems ". Unfortunately, however, problems have
become entrenched in all mathematical examinations, and
there is nothing for it but to teach boys how to solve them.
And, after all, problems do test boys' knowledge of certain
principles, and a correct solution is always a source of satis
faction.
The veriest tyro of a teacher can write out on the black
board the solution of a problem which the boys themselves
have been unable to solve. But what do the boys gain from
that? The mere setting out of a solution deductively, after
the manner of a proposition in Euclid, gives the boys no
inner light at all. The boys want to be initiated into a plan
of effective attack, to be taught how to analyse and how to
utilize the data of a problem, to be told exactly how the
teacher himself discovered the solution.
Be it remembered that a solution most suitable for a
class of boys is by no means necessarily the " neat " solution
so dear to the heart of a mathematician.
The main difficulty felt by boys in solving most algebra
problems is the translation of the words of the problem
into suitable equating formulae. Much practice is necessary
if facility in this translation is to be gained. Once expressed
in algebraic form, the equation is generally easy of solution,
The boy knows that an equation consists of two parts
connected together by the sign =. The first thing to search
for in a given problem is therefore the word " equal ", or
some words which imply " equal ", or such words as " greater
than " or " less than ". If the problem concerns money
matters, the boy may be able to dig out of his own know
ledge some relation of equality, e.g.
Cost price + profit = selling price;
216 CRAFTSMANSHIP IN MATHEMATICS
or, if he is dealing with racing problems, he may be able to
utilize the already familiar relation:
distance = speed X time;
or, if he is dealing with a clock sum, he may be able to split
up a component angle in two different ways, and so equate
a + P y ~f 8 (or some modification of this).
We append a few problems, with teaching hints.
1. If 4 be added to a certain number, and the sum be
multiplied by 5, the product will be equal to the number added
to 32. Find the number.
The question tells us,
a product the number + 32 (i)
Let us try to arrange our equation accordingly.
What have we to find? A number. Then let x represent
the number. The " product " is 5 times the sum of x and 4;
how shall we write this down? 5(# + 4). (i) tells us that
this product is equal to x + 32;
/. 5(# + 4) = x + 32. /. x = 3.
2. Find a number such that if it be multiplied by 5, and 2
be taken from the product, onehalf the remainder shall exceed
the number by 5.
The question says
half a remainder exceeds the number by 5,
i.e. half a remainder = the number \ 5 (i)
How shall we represent the number? by x. 5 times the
number? 5x. What is the remainder when 2 is taken from
this product? 6x 2. What is half this remainder? %(5x 2).
Then how from (i) can we make up our equation?
 2) = x + 5. .'. x = 4.
ALGEBRAIC EQUATIONS 217
3. A man spent 10 of his money, and afterwards one
quarter of the remainder. He had 30 left. How much had
he at first?
The word left suggests the relation:
(money at first) (expenditure) = 30. . . . (i)
Let us try to arrange our equation accordingly.
Let x represent the number of pounds he had at first. Then
x expenditure 30 (ii)
What is the expenditure?
First expenditure = 10; /. x 10 = remainder.
Second expenditure = } of remainder = J(*v 10),
.'. Total expenditure = 10 f \(x 10).
Now we may substitute this in (ii):
.'. x  {10 + \(x  10)}  30. /. X = 50.
4. A man buys a flock of sheep at 3 a head, and turns
them into a field to graze for 3 months, for which he is charged
45s. a score. He then sells them at 3, 10s. a head, and so
makes a clear profit of 77, 10s. How many sheep were there
in the flock?
Clearly:
(Money laid out) f (profit) (Proceeds of sale). . (i)
What have we to find? The number of sheep bought.
x
Let x represent no. of sheep bought; then = no. of
scores of sheep bought.
1. Money laid out:
(a) Cost of x sheep at 3 each = 3x pounds
(b) Cost of grazing  scores of sheep at 2 \ a score = f X 2
/. total money laid out = 3# f ( ~ X 2H . . . (ii)
2. Proceeds of sale:
Sale of x sheep at 3 J each = 3J# pounds. , . . (iii)
218 CRAFTSMANSHIP IN MATHEMATICS
According to (i), (ii) + ?7 = (iii),
i.e. 3* + ~ X 2i + 77J  3*. /. X  200
5. A boy was born in March. On the 18th of April he
was 5 times as many days old as the month of March was on
the day before his birth. Find his date of birth.*
This examination absurdity is simple enough, once the
wording is unravelled. Note that if a boy is born, say, on
4th May, he is 20 days old on 24th of May. It is a case of
simple subtraction.
In the problem we have to deal with two ages, expressed
in days:
(i) the age of the boy on J8th of April,
(ii) the age of March on the day before the boy was born
The former = 5 times the latter. Hence we can
make up our equation ........ (i)
(i) The age of the boy on 18th of April:
Let the boy be born on the #th of March.
By the end of March he is (31 x) days old.
By April 18th he is (31 * + 18) days old ..... (ii)
(ii) The age of March on the day before the boy was born.
The boy was born on the #th day of March.
Hence March was then x days old.
The day before that, March was (x 1) days old. . . (iii)
From (i) we know that (ii) is 5 times (iii).
i.e. (31  x) + 18 = 5(*  1). /. x = 9.
6. The 3 hands of a watch are all pivoted together centrally.
When first after 12.0 will the seconds hand, produced back
wards, bisect the angle between the other 2 hands?
We have to remember that the seconds hand moves 60
The problem is not well worded. For instance, March is not, strictly, nine
days old until midnight on March Qth. We have assumed that the boy was born
at midnight, and we have reckoned ages from midnight.
ALGEBRAIC EQUATIONS
219
times as fast as the minute hand, and the minute hand 12
times as fast as the hour hand. Thus the relative speeds are
720 : 12 : 1.
At noon (N) all the N
hands are together. The
watch circumference is
divided into 60 equal arcs,
and we may measure the
angles in terms of these
arcs. Let the seconds
hand move round to its
position S* in x seconds;
i.e. arc NS = x. Since
the minute hand also
moves round to its posi
tion M in x seconds, the
x
60'
And since the hour hand also moves round to its position
v
H, in x seconds, the arc NH measures  .
Now the seconds hand OS produced backwards, making
OS', bisects the angle HOM; i.e. the arc HS' = the arc S'M.
We ought therefore to be able to make up an equation
by means of the pieces of arc between N and M, e.g.
NM = NS' 4 S'M (i)
We know that NM = % ,
arc NM measures
Fig. 61
 NH) =
that NS' = x  30,
and that S'M = JHM 
.'. from (i) we have ~ = (x 30) + f ~ ~\,
:. x = 30^^ (sees, after 12.0).
The angles in the figure are necessarily much exaggerated.
220 CRAFTSMANSHIP IN MATHEMATICS
7. 54 minutes ago, it was 5 times as many minutes past 5
as it is now minutes to 7. What is the time now?
Most watch and clock problems can be solved on the basis
of the principle illustrated in the last example, and one
careful analysis, to exemplify the method, is usually enough
to enable the boys to attack successfully most of the problems
given in a textbook. But this problem, another absurdity
from an examination paper, does not fit into any general
scheme. Though easy, its translation into an equation may
at first puzzle most average pupils.
The basis for equalizing quantities is pretty obvious at
the outset:
(54 min. ago, no. of min. past 5) = 5 (no. of min. to 7 now). . (i)
The question to be answered is, what is the time nowl
The problem mentions 5.0 and 7.0, and refers to the time
now as a number of minutes to seven.
Hence, let the time now be x minutes to 7. . . . (ii)
We also require to know what the time was 54 minutes
ago\ this must have been (x + 54) minutes to 7.
But we have to express this time in terms of minutes
past 5.0. Now 5.0 is 120 minutes before 7.0.
Hence, 54 minutes ago the number of minutes past 5.0
was 120 (x f 54) (iii)
From (i), (iii) is 5 times (ii),
i.e. 120  (x  54) = 5*. /. x = 11.
All answers to equations should be checked; checking in a
case like this is particularly necessary.
Since x= 11, the time now is 11 minutes to 7, or 6.49.
The time 54 minutes ago was 5.55, or the number of minutes
(55) then past 5.0 is 5 times the number of minutes (11) now
to 7.0.
ALGEBRAIC EQUATIONS 221
8. Three friends going on a railway journey take with
them luggage amounting in all to 6 cwt. Each has more than
can be carried free, and the excess charged them is 2s. 6d., 7s.,
and 10s., respectively. Had the whole belonged to one person,
he would have had to pay 34s. 6d. excess. How much luggage
is each passenger allowed to carry free, what is the excess
charge per Ib., and what is the weight carried by each of the
three friends?
Consider first a simple case. If I am allowed to take
with me, say, 100 Ib. free, and have to pay, say, \d. on every
Ib. exceeding 100, then if I take with me a total of, say, 150 Ib.
the excess I have to pay is \d. X (150 100).
Thus a possible form of equation seems to be:
(excess charge per Ib.) X (no. of Ib. excess) = (total charge for excess),
and as there are two separate though similar statements
concerning excess, we ought to be able to formulate two
equations, say in x and y.
What have I to find out? (1) Ib. per person carried free,
and (2) excess charge per Ib. Hence:
Let each passenger
(1) Carry x Ib. free.
(2) Pay y pence on each Ib. excess.
Assume that one of the friends takes 3 tickets and shows
them to the porter, who on weighing the luggage and finding it
to be 672 Ib., deducts 3x from the 672, and charges y pence per
Ib. on the difference, viz. (672 3#). Since the sum actually
paid for excess = 2s. 6d. + 7*. + 10s. = 19s. 6d. = 234^., ,
/. (672  3x)y = 234 (i)
But if all 672 Ib. had belonged to one person, he would
have taken only 1 ticket, and the porter would have charged
y pence on each of (672 x) Ib. Since the sum actually paid
in this case for excess = 34s. Qd. = 414J.,
.'. (672  x)y = 414 (ii)
222 CRAFTSMANSHIP IN MATHEMATICS
Dividing (i) by (ii) we have
672  3x _ 13
672 * 23' /. #= 120
= no. of Ib. allowed free.
Hence from (i) or (ii),
y \d. = charge per Ib. excess.
Weight in Ibs. carried by each person
= 120 + ~, 120 + 8 * 120 + 1?, respectively.
Ill
This problem is worth giving a class a second time, say
three months after the first. Boys seem to find a first analysis
a little difficult.
9. An express train and an ordinary train travel from
London to Carlisle, a distance of 300 miles. The ordinary
train loses as much time in stoppages as it takes to travel 25
miles without stopping. The express train loses only three
tenths as much time in stoppages as the ordinary train , and it
also travels 20 miles an hour quicker. The total times of the
two trains on the journey are in the ratio 26 : 15. Find the
rate of each train.
We have to think of distance, speed, and time, and their
relation d = st.
The ratio of the total times taken by the two trains is
given, viz. 26:15 (i)
Hence if we can express these total times in some other way
we can formulate our equation.
Distance: 300 miles.
Speeds'. Let ordinary train travel x miles an hour.
Then express train travels x f 20 miles an hour.
Qf)f)
Times: Time taken by ordinary train = f stoppages (in
x
hours) (ii)
Time taken by express train =  + stoppages (in
X ~j~ 20
hours) . . (iii)
ALGEBRAIC EQUATIONS 223
We now require to know the amount of time lost over
stoppages.
(1) Ordinary train: time lost over stoppages is equal to
that taken in travelling 25 miles:
Train travels x miles in 1 hour,
= 1 mile in  hour
x
25
= 25 miles in hours
X
25
.*.  hours   time lost over stoppages.
x
(2) Express train: time lost over stoppages = ^ that of
ordinary train,
3 r 25 , 15 ,
_ of hours = hours.
10 x 2*
We can now express (ii) and (iii) in the following forms:
Time taken by ordinary train = ( .  ) hours. . (iv)
\ x x '
rr t t . / 300 . 15\ , , .
Time taken by express tram = I f J hours. (v)
\x f 20 2x'
Hence we have from (i), (iv), (v),
300 25
~x x 26
15 15
* + 20 2x
.'. X = 30 (miles an hour).
If in " racing " and analogous problems the relation d st
is kept in view, the necessary analysis is usually quite simple.
10. What is the price of sheep per 100 when 10 more in
100 worth lowers the price by 50 per 100?
We must avoid confusion between (1) the number of
sheep for 100 and (2) the cost of 100 sheep.
We can find the number of sheep costing 100 if we
know the price of 1, and we can find the price of 1 if we know
the price of 100.
224 CRAFTSMANSHIP IN MATHEMATICS
A possible equation seems to be:
(First no. of sheep for 100)
= (second no. of sheep for 100) 10. . . (i)
What have we to find? The price of 100 sheep.
(1) First price of 100 sheep. Call this #.
:. 1 sheep costs ^ ,
/. number obtainable for 100 = 122 = 10 ' 000 . . . (ii)
&L x
100
(2) Second price of 100 sheep. This = (# 50).
/. 1 sheep costs ( * " 50) .
.. number obtainable for 100 =
(# 50) x 50
100
(i) shows us how (ii) and (iii) are related, and then we may
make up our equation.
 10, ... * = 250.
x x 50
i.e. the price of 100 sheep is 250, or 2, 10*. each, or 100
worth 40 sheep.
(If 50 for 100, each costs 2, or cost of 100 = 200,
i.e. 50 less than before.)
I have found that even Sixth Form boys are sometimes
baffled by the analysis of this little problem.
Problems which are at all unusual in form are always
worth repeating after an interval.
Books to consult (on the general technique of teaching algebra):
1. The Teaching of Algebra, Nunn.
2. Elements of Algebra, 2 vols., Carson and Smith.
3. A New Algebra, Barnard and Child.
4. Algebra, Godfrey and Siddons.
6. A General Textbook of Elementary Algebra, Chapman.
6. Elements of Algebra, De Morgan. (A valuable old book. So
are De Morgan's other books, especially his Arithmetic.)
ELEMENTARY GEOMETRY 225
CHAPTER XX
Elementary Geometry
Early Work
Some of the younger generation of teachers have never
read Euclid, and seem to be totally unacquainted with the
rigorous logic of the old type of geometry lesson. Not a
few of the old generation regret the disappearance of Euclid,
urging that the advantages of the newer work are outbalanced
by the loss of the advantages of the older.
The real distinction between the older and the newer
work is, however, sometimes forgotten. Essentially, Euclid
wrote a book on logic, using elementary geometry as his
raw material. The amount of actual geometry, qua geometry,
which he taught was, relatively speaking, trifling. Boys in
existing technical schools do ten times as much geometry
as is found in Euclid. But as an exposition of deductive
reasoning from an accepted set of first principles, Euclid has
never been equalled.
Until forty years ago, Euclid was universally taught in
secondary schools, but the collective opinion of experts had
gradually hardened against it, partly because the average
boy found it difficult, partly because some of its propositions
were too subtle for schoolboys, partly because its foundations
were far from being unassailable, and partly because the
actual geometry it expounded was too slight to be of much
practical service.
But the geometry that was substituted for Euclid the
geometry now exemplified in all the ordinary school text
books is still Euclidean geometry, i.e. it is a geometry based,
in the main, on the same foundations as Euclid. These
foundations consist of a number of quite arbitrarily chosen
axioms. Other sets of axioms might be substituted for them,
and then we should get an entirely new system of geometry
(B291) 16
226 CRAFTSMANSHIP IN MATHEMATICS
of a nonEuclidean character. Reference to such geometry
will be made in a future chapter.
In practice, the difference between Euclid and the geo
metry now taught is in the choice of working tools. In Euclid,
the proof of every proposition was ultimately traceable to
the axioms, and every schoolboy had to substantiate every
statement he made by referring it to something already proved,
and this in its turn to something that had gone before, and
so back to the axioms. In those days the axioms were really
the working tools. But those axioms were so subtle that
the boys' confidence in them was entirely misplaced. Now
adays, the working tools consist of a small number of funda
mental propositions. By means of carefully selected forms
of practical work, the truth of these propositions is shown
to beginners to be highly probable, but the formal proofs
of such propositions are not considered until the boys reach
the Sixth Form. Examination authorities no longer call for
the formal proofs at the School Certificate stage.
These working tools once thoroughly mastered, beginners
plunge into the heart of the subject and make rapid headway.
In the old Euclidean days a year or more was spent on these
propositions and a few others, and at the end of that time
the average boy had but very vague notions about them,
though the mathematicallyminded boy certainly did seem
to appreciate the rigour of the reasoning presented to him.
I find it a little difficult to describe the methods of the
preeminently successful teacher of geometry. The methods
are not the reflection of any particular book but of the man
himself. By the gifted teacher who happens to be a sound
mathematician, a new principle is often illuminated by so
many sidelights that even the dullard can hardly fail to see
and understand. Successful teachers of geometry seem to
be those who have given special attention to the foundations
of the subject, who possess exceptional ingenuity in making
things clear, and who at an early stage make use of symmetry
and of proportion and similarity.
Those teachers who are not successful are often those
ELEMENTARY GEOMETRY 227
who confine their work to the limits of the ordinary text
book written for the use of boys; who fail to survey the whole
geometrical field; who are still unacquainted with, or at all
events do not teach, the great unifying principles of geometry
duality, continuity, symmetry, and so forth. The little
textbooks are all right for the boys, but the teaching of geo
metry connotes something outside and beyond such text
books, especially the principles underlying the grouping and
regrouping of the thousand and one facts that the beginner
necessarily learns as facts more or less isolated. The ac
cumulated facts can be given many different settings, each
setting forming a perfect picture, all the pictures different
yet closely related.
Work up to 13 or 14
As I have said in another place, 5 * the following is an
expression of authoritative opinion as to the nature of the
work which it is most advisable to do with young boys:
/
1. The main thing should be to give the boys an in
telligent (knowledge of the elementary facts of geometry A
2. No attempt should be made to develop the suoject
on rigorously deductive lines, from first principles, though,
right from the first, /precise reasons for statement^ made
should be demanded.
3. Young boys are never happy and are often suspicious
if they feel they are being asked to prove the obvious, but they
can follow a fairly long chain of reasoning if the facts are
clear.
4. All subtleties should be avoided, and, therefore, proofs
of propositions concerning angles at a point, parallels, and
congruent triangles should not be attempted, such proofs
being a matter for later treatment in the upper Forms.
5. These main working tools of geometry, angles at a
point, parallels, and congruent triangles, should be presented
* Lower and Middle Form Geometry, Preface.
228 CRAFTSMANSHIP IN MATHEMATICS
, gJ^
in such a way as to $nable the boys to understand them
clearly and to use and apply them readily.
6. Young boys can easily understand Pythagoras, ele
mentary facts about areas, and the main properties of the
circle and of polygons; and these facts should be taught.
7. The simple commensurable treatment of (i) the pro
portional division of lines, and (ii) similar triangles, should
be included in the work to be done at the age of 12 to 14;
young boys soon become expert in the useful practice of
writing down equated ratios from similar triangles.
8. By about the age of 13, a boy ought to be able to write
out a simple straightforward proof formally and to attack
easy riders.
9. Throughout the course, all possible use should be made
of the boys '( intuitions and of their knowledge of space
relations in practical life,) relations in three dimensions as
well as in two.
10. Responsible teachers should always express them
selves in exact (geometrical language, and should make pre
cision and accuracy of statement an essential) part of the
boys' training.
11. The boys should be taught how to formulate their
own definitions, and, under the guidance of the teacher, to
polish up these definitions as accurately as their knowledge
at that stage permits; and these definitions should be learnt.
Definitions should never be provided readymade.
12. The boys should be taught to realize exactly what
properties are implied by each definition, and all other pro
perties must be regarded as derivative properties requiring
proof.
13. The boys may usefully be given an elementary training
in the principle that a general figure necessarily retains its
basic properties even when it becomes more and more
particularized, but that, as the figure becomes less and less
general, it acquires more and more properties; and vice
versa.
14. In short, clear notions of the allimportant principle
ELEMENTARY GEOMETRY 229
of continuity should, by the time a boy is about 13, " be in
his very bones ".
15. A young boy's natural fondness for puzzles of all
kinds may often usefully be employed for furthering his
interest in geometry.
16. In one respect we have drifted too far away from
Euclid: boys' knowledge of geometry is too often vague,
too seldom exact.
Teachers differ in opinion about the degree of accuracy
to be demanded in beginners' geometrical drawing. Some
training in the careful use of instruments is certainly desirable,
but time should not be wasted over elaborate drawings
when freehand sketches can be made to serve adequately.
In Technical schools, accurate drawing with instruments is an
essential part of much of the pupils' work. Even in Secondary
schools, where the geometry is necessarily given an academic
bias, a preliminary training in the careful use of instruments
serves a useful purpose, but there is no point in making Secon
dary school boys spend time over elaborate pattern drawings
and designs. Exercises in accurate work of a more telling
type may be found in the theorems of Brianchon, Desargues,
Pascal, and others. As theorems, these are, of course, work
for the Upper Forms; as geometrical constructions, they
are useful in the lower Middle Forms, where they may be
learnt as useful and interesting geometrical facts.
All pupils should be taught the wisdom of drawing good
figures for rider solving purposes.
Boys in Technical schools often have a better allround
knowledge of geometry than those in Secondary schools
because they do more work in three dimensions. Solid
geometry of a simple kind may with great advantage be
included in the early stages of any geometry course.
A short course on simple projection at about the age of
13 helps later geometry enormously. Boys soon pick up the
main principles, and the work helps greatly to develop their
geometrical imagination. So does simple work with the
polyhedra, work which always appeals to boys.
230 CRAFTSMANSHIP IN MATHEMATICS
Let logic of the strictly formal kind wait until foundations
are well and truly laid. The increasing difficulty felt by
beginners in geometry is largely an affair of increasing diffi
culty of logic, and thus we now recognize that parts of the
third, fourth, and sixth books of Euclid are easier than parts
of the first book.
Push ahead. Do not paddle about year after year in the
little geometrical pond where examiners fish for their questions.
Even for the examination day such paddling most certainly
does not pay.
It is convenient, though not defensible, to preserve the
old distinction between axioms, postulates, and definitions.
But if any teacher still believes that axioms should be con
sidered in a beginner's course of geometry, let him consult
Mr. Bertrand Russell, and he will soon be disabused.
Early Lessons
Here are a few early lessons in geometry, selected at
random from the book already cited. The sections have
been renumbered, seriatim, for convenience of reference, but
actually the lessons are drawn from all parts of the book.
LESSON I
Planes and Perpendiculars
1. Carpenters, bricklayers, blacksmiths, and plumbers, all
have to know something about geometry. Architects, builders,
surveyors, and engineers have to know a great deal about it.
All of them have to know how to measure things, and how
to make things perfectly level, perfectly upright, perfectly
square, perfectly " true "; and much more besides.
You have already learnt how to use a ruler or scale,
marked with inches and parts of an inch on the one edge,
and with centimetres and parts of a centimetre on the other.
Note that there are very nearly, but not exactly, 2J centi
metres to an inch.
ELEMENTARY GEOMETRY
231
You know already that, when two lines meet at a point,
they form an angle. Here are three angles.
blade
Fig. 63
The middle one is the angle you know best. It is the angle
you see at the corner of an ordinary pictureframe, or of a
door, or of a windowframe, or of a tabletop. Such an angle
is called a square angle, or right angle.
2. You have probably seen a carpenter planing a piece
of wood, perhaps for a shelf. He begins
by planing one of the " faces " of the wood,
and, as soon as he thinks that the face is a
true plane, he tests it. To do this he uses
a try square, which consists of a steel
blade with parallel edges perfectly straight,
fixed at right angles into a wooden stock.
(A /rysquare is a tatfsquare.) He holds the stock in his
hand, and to the planed face of the wood he applies the
outside edge of the blade, " trying " it in many places and in
different directions, along and across. If he can see day
light anywhere between the blade and the wood, he knows
that the planed face is not yet a true plane, and that he
must continue his planing. When it is true, he marks it
face side.
3. Now he turns up the wood so that
an edge rests on the bench, and he planes
the edge at the top. Not only has he to
make this faceedge (as it is called) a true
plane like the faceside, but he has to make
the two planes at right angles to each other,
or, as we usually say, perpendicular to each other. The
carpenter is not satisfied until the inside right angle of his
trysquare fits exactly, at the same time, the faceside and
Fig. 64
232 CRAFTSMANSHIP IN MATHEMATICS
the faceedge, the blade fitting against the one, the stock
against the other, the test being made at several places along
the wood.
You might use a big trysquare to see if a flagstaff or a
telegraphpost is perpendicular to the ground. If it did not
fit exactly in the angle between the post and the' ground,
no matter where tested round the post, you would know
that either the ground is not level or the post is not upright.
Two planes, or two lines, or a line and a plane, are
perpendicular to each other if and only if they are
at right angles to each other. A perfectly upright post
in a sloping bank is not perpendicular to the bank because
it does not make right angles with the bank.
4. The maker of a trysquare guarantees the accuracy
of the inside angle, but not of the outside right angle. Thus
you may use it for testing the right angles
of a tabletop, of a door, of the outside of
a box. It is not advisable to use it for
testing the inside right angles of a box, or
i^g. 6 S of a drawer, or of a doorframe. (Strictly
speaking, it ought not to have been used
for testing the right angle round the flagstaff or telegraph
post.) For testing inside right angles, we use an architect's
setsquare, a flat triangular piece of wood with a true right
angle. You will be given two of these to work with, a little
later on.
5. You have learnt that, when you want to find out if a
surface is a true plane, you must test it with an accurately
made straightedge of some kind. If, on the other hand,
you are doubtful about the accuracy of a straightedge (an
ordinary ruler, for instance), you can test it by applying
it to a plane known to be true. Thus, a true straightedge
may be used for testing a plane, and a true plane may be
used for testing a straightedge. One must be true, and then
it may be used for testing the other.
(When numbered statements in dark type are followed by the
letter " L ", the statements are to be learnt, perfectly.)
ELEMENTARY GEOMETRY 233
6. A PLANE SURFACE (or a PLANE) is a surface
in which a true straightedge will everywhere fit
exactly. (L.)
LESSON II
Horizontal, Vertical, and Oblique Lines and Planes
7. Borrow a spiritlevel from the Geography Master,
or from the school carpenter, and see if the floor of your room,
the top of the table, the window ledge, and the mantelpiece
are horizontal (perfectly level). (Your master will explain
how the spiritlevel is made and used.) If a plane surface
is everywhere horizontal, the surface is called a horizontal
plane, and straight lines drawn on that surface are horizontal
lines. The surface of still water (in a basin, for instance) is
a horizontal plane, and floating leadpencils may be regarded as
representing horizontal lines. The edge of a bookshelf, the
edge of a table top, the joints of floorboards, the line where the
floor meets a wall, are other examples of horizontal lines.
8. You have probably seen a bricklayer use a plumb
line a cord stretched straight by a hanging leaden weight.
He uses it to see if the walls he is building are vertical
(perfectly upright). Make a plumbline for yourself, and see
if your school walls are vertical. If they are vertical and if
they are plane, their surfaces are vertical planes. Cover
the plumbline with chalk, hold it close to the wall and let
it come to rest, then pull it out towards you a little way and
let it go suddenly. It springs back and leaves a straight
chalkline on the wall. This straight line is a vertical line.
The balusters on a staircase, hanging chains, hanging
ropes, telegraph poles, the lines where any two walls of a
room meet each other, may all be regarded as representing
vertical lines. Raindrops fall in vertical lines, unless there
is a wind. A telegraph post fixed in horizontal ground is
both vertical and perpendicular; if fixed in a sloping bank,
it is vertical but not perpendicular. Why?
Vertical lines always point downwards, towards the
centre of the earth.
234 CRAFTSMANSHIP IN MATHEMATICS
Butterflies alight with their wings in vertical planes,
moths with their wings in horizontal planes.
9. Planes and straight lines which are neither vertical
nor horizontal are called oblique. Oblique means slanting
or sloping.
10. On the vertical surface of a wall, it is easy enough to
draw both horizontal and vertical and oblique lines, but
vertical and oblique lines cannot be drawn on a horizontal
sheet of paper lying on the table. All lines on a horizontal
plane are horizontal. Yet it would be inconvenient to have
to draw lines on a sheet of paper which is pinned to the
wall, though sometimes your master certainly does draw
lines on a vertical blackboard. It has been decided, just as
a matter of convenience when drawing, to represent the
three different kinds of straight lines all on a horizontal
plane, and in this way: horizontal lines, parallel to the top
and bottom edges of your paper; vertical lines, parallel
to the left and righthand edges of your paper; oblique
lines, lines in any other direction.
Horizontal lines Vertical lines Oblique lines
But remember that, as long as your paper is lying on the
horizontal table, it is not strictly true to say that the lines
you draw on it are anything but horizontal. We do not
obtain a true picture unless we hold the paper in a vertical
plane (against the wall, for instance). Then the vertical
lines may be made to appear really vertical.
11. HORIZONTAL PLANES are planes which are
perfectly level. (L.)
12. HORIZONTAL LINES are straight lines in a
horizontal plane. (L.)
13. VERTICAL PLANES are planes which are
perfectly upright. (L.)
ELEMENTARY GEOMETRY
235
Fig. 66
14. VERTICAL LINES are straight lines in a
vertical plane that point downwards towards the
centre of the earth. (L.)
15. Planes and straight lines which are neither
horizontal nor vertical are called OBLIQUE. (L.)
LESSON III
Solids and Surfaces
16. Here is a brick. Measure it. It is 9" long, 41" broad,
3" thick. It is a solid body, but, in geometry, we call it a
solid not because it is made throughout of a
particular kind of hard stuff but because it
occupies a certain amount of space. If the
brick were hollow and made of paper, we should
still call it, in our geometry lessons, a solid.
A room of a house is a solid; so is an empty box. Both have
length, breadth, and thickness. But we do not usually
speak of the thickness of a house or of a box. We say that a
house has length, breadth, and height, and a box length,
breadth, and depth. But all have three dimensions; that is,
we can measure them from front to back, from side to side,
and from top to bottom. (Both the word dimension and the
word mensuration are derived from
the same Latin word, mensura, a
measure.)
17. Here are a cube (fig. 67, i)
and a square prism (ii). You
have probably seen them before,
and know their
names. If they
were made of paper,
we could run a
knife along some of
their edges and lay
them out flat like
this: Fig. 68
Fig. 67
236
CRAFTSMANSHIP IN MATHEMATICS
Plans of this kind are called the nets of the solids. Later
on in the geometry book, you will find instructions how to cut
out nets from stiff paper and how to fold and bind them
up into the solids they represent.
18. The surface of both the cube and the prism consists
of six faces. All six faces of the cube are squares. Only
two faces of the prism are squares, the other four being
oblongs. Sometimes we speak of the two square faces of
the prism as ends or bases, and the four oblong faces as
the sides. In each case, all the faces are, of course, planes.
Any two adjoining planes of the cube or of the prism meet
in an edge, or, as we sometimes say in geometry, the two
planes intersect in a straight line.
19. Here are four more solids which you have probably
seen before; a square pyramid, a cylinder, a cone, and
a sphere.
Fig. 69
20. In shape, the square pyramid reminds you of the
famous Egyptian pyramid. Its surface consists of five plane
faces, namely, one square base, and four triangular faces
meeting in a point called the vertex. The vertex is exactly
over the centre of the base * (fig. 69, i).
21. The cylinder reminds you of a garden roller, of a
jamjar, or of part of a pipe or tube. You can imagine it
spinning on an axis. When rolled on the ground it runs
in a straight line. The complete surface of the cylinder
consists of two circular plane surfaces separated by a curved
surface (fig. 69, ii).
22. The cone reminds you of the oldfashioned candle
* It is convenient to be able to refer to the " centre " of a square, but it is not
strictly correct. A circle has a true centre, so has a sphere.
ELEMENTARY GEOMETRY
237
extinguisher, or of the sugar loaf. It has a circular base,
and it is so far like a pyramid that it has a vertex over the
centre of the base. You can imagine it spinning on an axis.
When rolled on the ground it runs round in a circle. The
complete surface of the cone consists of one circular plane
surface and a curved surface (fig. 69, iii).
23. The sphere reminds you of a ball of some kind,
and it is a ball which is perfect in this way the point called
the centre is exactly the same distance from every point
on the surface. You can imagine it spinning on an axis,
like the earth. When rolled on the ground, it will run in
any direction. The surface of a sphere is everywhere curved
(fig. 69, iv).
24. When we speak of a " cylindrical surface " or of a
" conical surface ", we usually refer to only the curved surface
of the cylinder or cone. It is important to notice that this
curved surface of the cylinder and the cone is very different
from the curved surface of a sphere. If you place a sphere
upon a plane (say a table), it touches the plane in a point.
If you allow a cylinder or a cone to lie with its curved surface
on a plane, it touches the plane in a line.
25. We can make nets of a cylinder and a cone, but not
of the sphere. Here arc nets of a square pyramid, a cylinder,
and a cone, but to make models from the nets of the last
two is a little difficult.
26. You have learnt to recognize a square, an oblong,
a triangle, and a circle. All these are called plane figures,
238 CRAFTSMANSHIP IN MATHEMATICS
because each encloses, within a boundary line, part of a
plane surface. All plane figures have closed boundary
lines. The letter O and the letter D are geometrical figures,
but not the letter C or the letter W. Straightlined figures
like squares and triangles are called rectilineal figures.
A circle is a curved figure.
27. A PLANE FIGURE is part of a Plane, and it
is separated from the rest of the Plane by a boundary
line. (L.)
28. A PLANE RECTILINEAL FIGURE is a straight
lined figure on a Plane. (L.)
29. RECTILINEAL means straightlined. (L.)
LESSON IV
Angles
30. If I stand facing the east and the drill sergeant says
" left turn ", I turn and face the north, and I have then
turned through a right angle. If he repeats
the order, I turn to the west, and I have then
turned through another right angle. If he
repeats the order twice more, I turn and face
south and then turn and face east, by which
time I shall have turned through four right
angles. I have made one complete rotation
(Lat. rota = a wheel). Note the little arrow showing my
first quarterrotation or right angle.
Evidently an angle may be smaller or greater than a right
angle. Whenever you look at a clock, the two hands are
making an angle with each other. In fact, they are making
new angles with each other all day long. Even when they
are exactly together they have just completed a new angle
and are just beginning to make others.
31. On paper an angle is represented by two lines meeting
in a point. The two lines are called the arms of the angle,
and the point where the two arms meet is called the vertex
w
ELEMENTARY GEOMETRY 239
of the angle. The same angle may have long arms or short
arms. If a big clock and a little watch are both keeping
correct time, the angles between their hands are always
exactly the same. An angle always represents an amount
of movement, namely, the movement of rotation. One
arm shows where the rotation began, and the other where it
finished, and you must always think of an angle in this way.
32. You can measure angles of different sizes fairly well
by opening and closing your dividers, but the joint prevents
you from making an angle of a whole rotation. A more
convenient form of angle measurer is necessary, and you
may make one in this way. Take two nicely planed strips
of wood, say about 12" long, " wide, J" thick, and pivot
them together, something like your dividers, by means of
a tiny brass bolt with rounded head and nut, generally
obtainable for a penny or two from the ironmonger's. If
you cannot obtain these things, two strips of cardboard
will do, pivoted on a long drawingpin, head downwards,
with a protecting bit of cork over the point.
Place your anglemeasurer on the table before you, the
vertex O to the left, the two arms OA, OB together as if
A,B
fcj
.L,
Fig. 72
they were both pointing to III on a clockface (fig. 72, i).
Keep the under arm OA fixed, and rotate the upper arm
OB. Rotate it in an anticlockwise direction (this is the custom
in geometry), and make angles equal to one, two, three, and
four right angles. Draw the four angles, and in each case
show the amount of rotation by means of little curved arrows
(fig. 72, ii, iii, iv, v).
33. In measuring different quantities, weights and mea
sures for instance, big units like tons and miles are some
240 CRAFTSMANSHIP IN MATHEMATICS
times inconvenient. We do not weigh our tea in tons or
measure our pencils in miles; we use smaller units like
ounces and inches. So with angles. A right angle is a rather
00* big unit, and sometimes we use a
smaller unit called a degree. If we
make a right angle as before, but move
OB into position gradually, in ninety
equal steps, each of these steps is an
angle of one degree. It is a very
small angle, too small to be shown
clearly on paper unless we make the
n g , 73 arms very long. The figure shows that
even an angle of 5 degrees is very small.
The sign for " degree " is a little circle placed at the top
righthand corner of the given number. Thus for " 35
degrees " we write, " 35 ".
34. We may make up a little table:
90 degrees make a right angle,
2 right angles make a straight angle,
2 straight angles make a perigon.
A perigon is an angle of one complete rotation (pert round;
gon ~ angle). It is equal to four right angles, or 360. A
straight angle contains 180 ( 32, fig. 72, iii).
We choose the number 360 for the perigon simply because
it is the number which contains many useful factors (2, 3,
4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72,
90, 120, 180). Any other number would do, but it would be
less useful. The French use the number 400; they prefer
to divide up the right angle into 100 degrees (they call them
grades) instead of 90.
35. Note the number of degrees in the angles of fig. 74.
The dotted lines show the right angles, and help the eye
to estimate the numbers of degrees.
Practise drawing angles of different sizes, and estimating
the number of degrees. The most important angles of all
are 30, 45, 60, 90, 180. The easiest to make is, of course
ELEMENTARY GEOMETRY
241
an angle of 90. Divide it into two equal parts as accurately
as you can, and so obtain 45. It is pretty easy to divide,
iB
Fig. 74
with fair accuracy, 45 into three equal parts, in order to
obtain 15 and 30. And so on.
But guesswork will certainly not always do. It is often
necessary to draw given angles accurately, and for this pur
pose you must use a pro
tractor. A protractor is
a semicircular * piece of
brass or celluloid, with
numbers from to 180
round the circumference,
in both directions, and by
means of it you can make Fig. 75
an angle of any size.
Suppose you have to draw a line, at say, 55 with a given
line. Place the straight diameter of your protractor against
the given line, in such a way that the marked midpoint
*A surveyor's protractor is circular, and is numbered up t
geometry we do not very often require angles greater than ioo.
(B291)
to 360. But in
17
242 CRAFTSMANSHIP IN MATHEMATICS
of the diameter is against that end of the line that is
to be the vertex of the angle. At the number 55 on the
circumference, mark a point on the paper. Remove the
protractor, and through that point draw the second arm of
the angle.
But, you will say, there are two 55's on the circumference.
How are we to choose between them? That is easy, for
you know that 55 is less than a right angle, and you choose
the 55 which will give you such an angle. The other 55
would be used if the vertex of the angle had to be at the
other end of the line.
36. When two straight lines stretch out from one
point, like two spokes from the hub of a wheel, they
form an angle. (L.)
37. The two lines are called the arms of the angle,
and the point where they meet is called the vertex
of the angle. (L.)
38. An angle always shows a certain amount of
ROTATION round the vertex, one arm showing
where the rotation began, the other arm showing where
it ended. (L.)
39. A PERIGON is an angle of one complete
rotation. (L.)
40. A STRAIGHT ANGLE is an angle of a half
rotation. (L.)
41. A RIGHT ANGLE is an angle of a quarter
rotation. (L.)
42. AN ANGLE OF ONE DEGREE is an angle of
sio P art * a rotation.
LESSON V
Surveyors and their Work
43. A surveyor's work is to measure up land, and to
draw plans and maps. For measuring lengths, he uses a
long chain of 100 links. For measuring angles, he uses an
ELEMENTARY GEOMETRY 243
anglemeasurer which is like yours in this respect that
it consists of two pivoted arms; but it is much more elaborate
than yours, for he has to measure angles very accurately.
He also uses a levellingstaff, to help him measure differences
of level. A levellingstaff is merely a pole, graduated to show
heights above the ground.
44. Here is a problem in which the necessary angle
measurements may be correctly and easily made with one
of your setsquares. To solve it you must make a drawing
to scale.
A and B are two towns 20 miles apart. Another town C
is 60 east of north from A and 30
west of north from B. Draw a plan to
show the position of C, and give its
distances from A and B.
The line AB is 20 miles long,
and we have to draw it to a suitable
scale. A scale of " to the mile would
do. Thus we make AB 20 eighthinches, or 2", long.
If C were exactly north of A, it would be somewhere
in the line AD. But it is 60 east of this line, and we there
fore make the angle DAF equal to 60. Again, if C were
exactly north of B, it would be somewhere in the line BE.
But it is 30 west of this line, and we therefore make the
angle EBG equal to 30. We know now that the town C
lies on both AF and BG. But the only place where it can
lie on both is where they meet. Hence, mark this point, C.
We have thus found the position of C.
To find the distances CA and CB, we measure them to
scale. CA is nearly 17J eighthinches long, and CB is 10
eighthinches long. Thus C is 17 miles from A and 10
miles from B.
But this problem was a problem on paper. No part of
the work was done with measuring instruments in the
field. Let us come back to the surveyor.
Sometimes a surveyor works on level ground, and has to
measure angles in a horizontal plane. Sometimes he works
244
CRAFTSMANSHIP IN MATHEMATICS
on hilly ground and has to measure angles in a vertical plane.
45. Measuring an angle in a horizontal plane.
Suppose you are standing at a place P in a field, and you
imagine a line drawn from your eye to each of two distant
trees, T l and T 2 . What is the angle between the lines? Set
up a table at P, with a piece of drawingpaper pinned on
it. (A camera tripod stand with a drawingboard fixed on it
horizontally about the height of your top waistcoat button
would do nicely.) Place an anglemeasurer on the table,
swing one arm round to point to T x , and the other round
to point to T 2 . Hold the arms firmly and draw the two
angle lines (against the inside of the arms), remove the measurer,
and with your protractor find the number of degrees in the
angle.
(Ask the geography master to show you his planetable
and to explain how he measures angles made by distant
objects. With his anglemeasurer pivoted to the centre of a
circular protractor on the table, he is able to read at once
any angle made by the two arms.)
46. Measuring an angle in a vertical plane. Pivot
your anglemeasurer to the side of a short post, or to the
side of a stout stick thrust
vertically into the ground, in
order that the arms may swing
in a vertical plane. An ordi
nary drawingpin makes a poor
pivot, for it is then difficult to
make the arms remain in a
particular position. An angle
measurer made of wood, with
a fairly tight wooden or metal
pivot, is much more satisfactory
than the pivoted cardboard
strips. An angle measured
in a vertical plane is always an angle with a hori
zontal arm; the other arm points upwards or downwards
as may be necessary.
Angle of elevation Angle of depression
Fig. 77
ELEMENTARY GEOMETRY 245
If you are on low ground and want to measure the angle
made by, say, a cottage at the top of the hill, point the one
arm of your anglemeasurer upwards to the cottage, and
measure the angle of elevation. If you are on high ground,
say the top of a cliff, and want to measure the angle made
by a boat in the water below, point the one arm of your
anglemeasurer downwards to the boat, and measure the
angle of depression. Since, for measuring different angles,
the arms may have to swing round in different vertical planes,
it is an advantage to be able to turn the post round in the
ground, and it should therefore have a rounded point, some
thing like the point of a cricketstump, prolonged.
(Ask the geography master to show you his clinometer, and
to explain how he reads, from the cardboard protractor,
angles of elevation and depression. Try to understand the
use of the little plumbline, and observe the pivot on which
the protractor turns.)
47. How can I find the width of a river which I
cannot cross? To solve this problem we have to measure
angles in a horizontal plane. E
Let AB and CD repre A ~
sent the two banks. I note
some object E on the oppo C
site bank, and I measure Fig. 78
any length FG, say 100
yards, on the near bank. Then I measure the horizontal
angles at F and G, in each case pointing one arm of my angle
measurer along CD and the other arm to the object E. I
note that angle EFG = 60, and angle EGF = 45. Now I
am ready to make a drawing to scale. A scale of 1" to
50 yards seems convenient, so that FG = 2". The width
of the river is shown by a perpendicular EH drawn from
E to FG. Measuring EH to scale, I find it is very nearly
yards.
48. How can I find the height of the flagstaff in the
school field? To solve this problem I have to measure an
angle in a vertical plane, and as one arm of my anglemeasurer
246
CRAFTSMANSHIP IN MATHEMATICS
will have to point upwards, the angle will be an angle of
elevation.
Let AB be the flagstaff, and let CD be the post to which
my anglemeasurer is attached: a convenient height of this
attachment is 4' above the ground. The post may be fixed
at any measured distance from the
flagstaff, say 20'. Thus DB = 20'.
The horizontal arm of the angle
measurer points to E in the flagstaff;
E is therefore 4' above the ground.
The other arm points to the top of
the flagstaff. I now measure the angle
ACE, and find it is 60. Now I am
ready to make a drawing to scale, say
1" to 10', so that CE (= DB) = 2",
angle ACE  60, angle AEC = 90.
The length of AE, measured to scale,
is 346'. Hence AB = 346' + 4'
= 386'.
(The length BE is exaggerated in
the printed figure.)
49. The next time you see a surveyor at work, ask him
to show you the instruments he uses for measuring horizontal
and vertical angles, and to explain how he is able to measure
even very small fractions of a degree. Also ask him to tell
you something about his levellingstaff and his chain.
LESSON VI
Symmetry
50. Stand in front of a lookingglass, with a book (or
some other object) in your right hand. In the glass you see
an image of yourself, but the image holds the book in his left
hand. Close your left eye; the image closes his right eye.
Hold open your right hand in front of the glass, and look
at the image of the palm. Compare this image with the
ELEMENTARY GEOMETRY 247
palm of your real left hand. They are exactly alike. For
instance, the two thumbs point in the same direction.
Thus the image of your right hand is a left hand.
In short, your two hands are not in all respects alike; each
is the " image " of the other.
51. Place a pair of gloves side by side on the table,
backs upwards, thumbs touching. Each is the image of the
other. Turn the left glove inside out; it has become a right
hand glove. You now have two righthand gloves, no longer
images of each other but like each other.
52. Fold a sheet of white paper, like a sheet of note
paper, and smooth down the crease. Open again, and let
a drop of ink fall in the crease. Now fold, and press the
folded paper fairly hard, to make the ink run and form a
pattern. Open; the two halfpatterns are right and left
handed; each is the image of the other. When the paper
is folded on its crease and held up to the light, the two half
patterns are seen to fit over each other exactly.
53. Right and lefthanded patterns that can be folded
exactly together in this way, and are thus images of each
other, are said to be symmetrical. The dividing line re
presented by the crease is called the axis of symmetry.
We say that the doubled pattern is symmetrical with respect
to the axis.
54. Take another sheet of paper, and fold as before.
Let a drop of ink fall inside, but at some distance from the
crease. Press down the doubled paper, and
so form ink figures. The figures are images of
each other and fold together exactly as before,
but this time they do not touch the crease (the
axis). That does not matter. They are still
symmetrical with respect to the axis.
55. In the accompanying figure (a kite),
the line AB is evidently an axis of symmetry, FI. so
for the half ACB can be folded over on AB
and be made to fit exactly on the other half. The one half
is the image of the other. Hence corresponding lines on
248 CRAFTSMANSHIP IN MATHEMATICS
the left and right must be equal in length. If we hold
the doubled paper up to the light, we can see that the lines
are equal. Corresponding angles must also be equal.
56. ABC and DBF are two figures symmetrical with
respect to the axis MN. Hence, if we fold on MN, the
figures will fit together exactly. On
ABC, mark the two points G and
H, fold over, and prick through G
and H, on DEF. On opening out
we shall find the two points K and
L in positions corresponding ex
actly to G and H, and KL is
evidently equal to GH.
57. Every point, every line, every
Fig. si angle, in one of two symmetrical
figures has an image in the other.
The image always corresponds exactly to the original. The
two have exactly corresponding positions.
58. Thus a point and its image are always equidistant
from the axis. In the last figure, for instance, A and D are
equidistant from MN, for they come together when the
figures are folded about the axis. Hence if we join AD, the
axis must bisect AD.
59. Fold a piece of paper and mark the
crease as an axis AB. On one side of the axis,
make a point M. Fold, and prick through M
to obtain its image N. Join MN. PM = PN
(by 58). Angle APM = angle APN (by
57). But angle MPN is a straight angle, and
thus the equal angles APM and APN are both
right angles. Also, the angles vertically opposite
Fig. 82 these are equal. Hence all four angles at P are
right angles. We see now that the axis not
only bisects MN but is perpendicular to it, that is, the axis
is the perpendicular bisector of MN. Observe that
whenever you fold a sheet of paper a second time, as when
you put it into an envelope, you make two axes of symmetry
M
ELEMENTARY GEOMETRY
249
perpendicular to each other, the four perfect right angles
fitting exactly together in the envelope. When the paper
is opened out, you see the complete perigon they form.
60. Fold a piece of paper, and then fold a second time,
thus obtaining two axes of symmetry, MN and PQ, and
four right angles. The four divisions are sometimes called
quadrants. Prick through all four thicknesses of the folded
paper, in four or five points not in the same straight line.
Open out, and join up the
points, in the same manner, in
the four quadrants, thus mak
ing four figures. Convince
yourself that both MN and PQ
are really axes, by first folding
on MN, holding up to the light
and seeing that the left and
right halves of the whole fit,
then folding on PQ and seeing
that the upper and lower halves
fit. Join any point, say A, to its image on the other side
of each axis, namely to B on the other side of MN, and to
C on the other side of PQ. Observe that the axes are the
perpendicular bisectors of the respective joining lines, MN
of AB, and PQ of AC. So it is generally.
61. When two figures are symmetrical with respect
to an axis, they are right and lefthanded, and when
they are folded about the axis, they fit together exactly.
(L.)
62. An axis of symmetry is the perpendicular
bisector of the line joining any point on one side of
the axis to its image on the other side.
250 CRAFTSMANSHIP IN MATHEMATICS
LESSON VII
Congruent, Symmetrical, Similar
63. We often require a word to describe two figures
which are alike in all respects corresponding lines the
same length, corresponding angles the same, areas the same,
appearances the same.
When two figures are exactly alike in all respects, and
can be made to fit exactly together, they are said to be con
gruent. (Congruent means exactly agreeing.) Here are
three pairs of congruent figures.
Fig. 84
64. Symmetrical figures are exactly alike in all respects
save one: they are right and lefthanded. To make two
symmetrical figures fit exactly together, we have to turn
one of them over through a straight angle (180), round
the axis of symmetry. It is like picking one up, turning it
upside down, and putting it down again. Then the two
will fit exactly.
Strictly speaking, we ought not to call symmetrical
figures congruent, because they are not alike in all respects;
they are right and lefthanded. But it has become customary
to call even symmetrical figures congruent, because they can
be made to fit exactly if one is turned over.
i. Congruent ii. Symmetrical iii. Similar
(but not symmetrical) (and congruent) (neither congruent nor
symmetrical)
Fig. 85
65. But similar is another term altogether. Similar
ELEMENTARY GEOMETRY 251
figures are figures of the same appearance, irrespective
of their size. (See fig. 85.)
66. Congruent figures which are not symmetrical may
be made to fit together exactly by sliding one over the other.
But symmetrical figures cannot be made to fit by sliding;
one has first to be turned over.
You might say that since congruent figures are alike
in appearance, we might call them similar. That is true,
but in geometry we do not usually apply the term similar
to congruent figures unless they are of different sizes.
67. Notice two important things about similar
figures: (1) all the angles in the one are equal to the
corresponding angles in the other; (2) the proportions in
the one are equal to the proportions in the other. (If, for
instance, the big pig's tail is onethird the length of his
back, the little pig's tail is onethird the length of his back.)
You will learn more about " proportions " later on.
68. CONGRUENT figures are figures exactly alike
in all respects. One can be made to slide over the
other and fit. (L.)
69. SYMMETRICAL figures are right and left
handed congruent figures. To make them fit, one has
to be turned over through 180, (L.)
70. SIMILAR figures are figures of different sizes,
but they have the same appearance, the same pro
portions, and the same angles. (L.)
LESSON VIII
Classifying and Defining
71. When we arrange a number of things in separate
classes, we are said to classify them.
We may, for instance, arrange all school exercisebooks
in two quite distinct classes, namely, ruled and unruled.
Such a classification is good. But suppose we say that
all the people in London are either males, or females, or
252 CRAFTSMANSHIP IN MATHEMATICS
Australians. The classification is bad, for the Australians have
been included twice over; they are all males or females.
Here is another example of a good classification. In a
certain school, the 100 boys in Form IV are grouped in
four divisions, according to the languages they learn in
addition to English and French.
Form IVa learn both Latin and Greek, but not German.
Form IVb learn both Latin and German, but not Greek.
Form IVc learn Latin, but not Greek or German.
Form IVd learn German, but not Latin or Greek.
There are four distinct divisions. Every boy is included
once, and only once.
72. Now we will classify triangles. All triangles are
either isosceles or scalene. But isosceles triangles are of two
kinds, those with two sides equal, those with all three sides
equal. Thus we may arrange the classes in this way:
Triangles
no two sides equal at least two sides equal
I
I I
only two sides equal all three sides equal
or we may arrange in this way:
Triangles
scalene isosceles
I I
base shorter or longer base equal to the equal sides
than the equal sides equilateral
73. Defining. A thing (it may be a dog or it may
be a triangle) has a name, and that name is a word. In
order to say what that word means, we have to make a
short statement which will show how the thing is dis
ELEMENTARY GEOMETRY 253
tinguished from all other things. That short statement is a
definition. We define a word, and the definition must
include the leading property of the thing.
We begin by thinking of the class to which the thing
belongs. Suppose, for instance, we have to define a chair.
To what class of things does a chair belong? Evidently
to the class articles of furniture. Thus we may begin
by saying,
A chair is an article of furniture . . .
Now we have to pick out the particular property which
distinguishes a chair from all other articles of furniture.
What is a chair specially used for? For sitting on. Thus
we may now say,
A chair is an article of furniture for sitting on.
But benches, sofas, and stools are also used for sitting on.
How are we to distinguish a chair from these? Benches
and sofas are made for more than one person to sit on. So
we may say,
A chair is an article of furniture for one person to sit on.
But this might apply to a stool. How are we to distinguish?
A chair has a back, a stool has not. We therefore say,
A chair is an article of furniture for one person to sit on
and to lean back against.
Again, define a pair of compasses. To what class
does it belong? Mathematical instruments. What is its special
use? For drawing circles. Thus we make up the definition:
A pair of compasses is a mathematical instrument for drawing
circles.
74. Define a triangle.
To what class does it belong? Plane rectilineal figures.
What property distinguishes it from all other plane
rectilineal figures? It has three sides. Therefore we say,
A triangle is a plane rectilineal figure with three sides.
254 CRAFTSMANSHIP IN MATHEMATICS
Define an isosceles triangle.
To what class does it belong? Triangles.
What distinguishes isosceles triangles from the other
great class of triangles (scalene)? Equality of the two sides
from the vertex to the base.
Therefore we say,
An isosceles triangle is a triangle in which the two sides
from the vertex to the base are equal.
Define an equilateral triangle.
To what class does it. belong? Isosceles triangles.
What distinguishes it from other isosceles triangles? The
base is equal to each of the other two sides.
Therefore we say,
An equilateral triangle is an isosceles triangle in which
the base is equal to each of the other two sides.
Another definition of an equilateral triangle is some
times given: an equilateral triangle is a triangle with three
equal sides. But this definition is not so good as the other.
75. We might, if we liked, classify triangles according
to their angles, and ignore their sides. The sum of the three
angles of a triangle is 180. Hence, if a triangle has an obtuse
angle, the other two angles must be acute; or if it has a right
angle, the other two angles must be acute; if it has neither
an obtuse angle nor a right angle, all three angles must be
acute. Thus we have a new classification: All triangles
are either obtuseangled triangles, or rightangled
triangles, or acute angled triangles.
But do not mix up the two classifications of triangles.
That would take us back to the Australians!
Such lessons are easily within the range of very young
boys.
Some teachers are, however, curiously afraid of the prin
ciple of symmetry, urging that it does not lend itself to
ELEMENTARY GEOMETRY 255
strictly deductive proof. Personally I would use it very
much more for teaching even advanced geometry; I always
did in my teaching days. For elementary work at all events,
it is a singularly useful weapon. Though proof by means
of it is difficult for beginners to set out, it produces con
viction in the beginner, a great gain.
It will be observed that, for framing definitions, we have
used the old device per genus et differentiam. This is probably
the only safe method for beginners. From schoolboys we
must be satisfied with something much less than perfection
in their definitions. In particular, do not worry about " re
dundant " definitions. In the early stages they are inevitable;
they are then almost to be encouraged. It is much better
to let a young boy say that " a rectangle is a rightangled
parallelogram " than " a rectangle is a parallelogram with
a right angle ". A beginner naturally,., regards the latter
with suspicion. It is doubtful wisdom ever to ask a boy to
define a straight line or an angle. He has clear notions of
these things already, and these notions he cannot express
in language that is entirely satisfactory. If a boy says that
" a straight line is the shortest distance between two points ",
strictly the definition is unacceptable, because of the vague
term " distance ". If he adds as tested by a stretched string,
we should feel that the idea in his mind was clear and distinct;
and what more can we want from him? As for an angle, I
have often asked boys for a definition, not because I expected
a satisfactory one, but in order to show them that, whatever
definition they put forward, it was open to criticism. Who
has ever defined either a straight line or an angle satisfactorily?
Again: ordinarily we distinguish between a circle and its
circumference, and a useful distinction it is. And yet we
all talk about drawing a circle to pass through three points.
However, matters of this kind are not for beginners but for
the Sixth Form, which is the proper place for a final polishing
up of all such things.
256 CRAFTSMANSHIP IN MATHEMATICS
Working Tools for Future Deductive Treatment
These consist of the familiar propositions concerning:
1. Angles at a point.
2. Parallels.
3. Congruency.
4. Pythagoras.^
5. Circles; such properties as can be established
from considerations of symmetry.
The formal proof of Pythagoras is easily mastered in the
Fourth Form, but proofs of the other theorems may wait
until the Sixth. Meanwhile all the propositions must be
thoroughly known as geometrical facts, facts which can readily
be used and referred to in all subsequent work. Although
formal proofs are beyond beginners, the probable truth of
the propositions must be substantiated in some way. Justi
fication is always possible at this stage, though rigorous proof
is not. Most of the more recent textbooks provide " practical "
proofs of a kind which to the beginner really do seem to
justify the claims made by the theorems. Here, little need
be said about such proofs.
First considerations of angles at a point naturally arise
when the nature of an angle itself is being discussed. Acute,
obtuse, adjacent, reflex, complementary, supplementary,
and vertically opposite angles may all be brought into an
early lesson, provided that the rotational idea of the angle
is clearly demonstrated. Angles up to 360 should be con
sidered from the first.
Here is a first lesson on parallels and transversals.
Parallel Lines and Transversals
76. You have already learnt that the blue lines on the
pages of your exercise books are parallel, that is, they run
in the same direction and are always the same distance
apart. When we speak of " distance apart " we mean the
ELEMENTARY GEOMETRY 257
shortest distance, and that distance is represented by a
perpendicular from one line to the other. But can we be quite
sure that a line which is perpendicular to one of the parallel
lines is also perpendicular to the other?
77. A line that is drawn across two or more other lines
is called a transversal (trans means across). Draw a trans
versal PQ across the parallel lines
AB and CD, cutting AB in M and p \
CD in N. A
Imagine a man to walk along AB
and, on reaching M, to turn to the c
right and walk along MN. He has
turned through the angle a x , for Fig. 86
he was first walking towards B,
and is now walking towards Q. On reaching N, let him
turn to the left, and walk along ND. He has now turned
through the angle a 2 , for he was walking towards Q and
is now walking towards D. But now that he is walking
along ND he is walking in the same direction as when
he was walking along MB. Hence the angle he turned
through on reaching N is equal to the angle he turned through
on reaching M, that is, the angle a 2 is equal to the angle a v
We might have expected this, for the two angles a x and
a 2 look alike. They are called corresponding angles.
78. When a transversal is drawn across two
parallel lines, the corresponding angles are equal.
(L.) Hence,
79. A transversal which is perpendicular to one
of two parallel lines is also perpendicular to the other.
(L.) Conversely,
80. If two lines are both perpendicular to a
transversal, they are parallel to
each other. (L.) A
81. Just as we showed that the
corresponding angles c^ and a 2 are c
equal, so we may show that the
corresponding angles fa and fa are Fig . 87
(E291) 18
258 CRAFTSMANSHIP IN MATHEMATICS
equal; also a 3 and a 4 ; also j8 3 and /? 4 . (Fig. 87.) But a 2 and a 4
are also equal, because they are vertically opposite angles.
Since a t = a 2 , ( 78)
and since a 2 = a 4 ,
therefore a A = a 4 .
The angles aj and a 4 are on opposite sides of the transversal,
and are called alternate angles.
Similarly it can be shown that the alternate angles fi 3
and /9 2 are equal.
82. When a transversal is drawn across two
parallel lines, the alternate angles are equal. (L.)
83. Observe that, in the eight marked angles of the
last figure, there are four pairs of opposite angles, four pairs
of corresponding angles, two pairs of alternate angles, every
pair being equal. The four a's are equal, and the four fi's
are equal.
The four angles between the parallel lines are called
interior angles.
The four angles outside the parallel lines are called
exterior angles.
The following is very important:
a i + Pa a straight angle,
two right angles.
But p 3  p 2 , ( 82)
therefore ^1+^2 two right angles.
Similarly we may show that a 4 + /? 3 = two right angles.
84. When a transversal is drawn across two
parallel lines, the two interior angles on the same
side of it are together equal to two right angles, that
is, they are supplementary. (L.)
Considerations of congruency are best led up to by actual
practical work on the construction of triangles from given
data. One lesson is enough for the boys to discover that a
triangle can be described if
ELEMENTARY GEOMETRY
259
(1) the 3 sides are given,
(2) 2 sides and the included angle are given,
(3) 1 side and 2 angles are given;
md that therefore two triangles are congruent if there is
:orrespondence and equality of
(1) 3 sides,
(2) 2 sides and the included angle,
(3) 1 side and 2 angles.
Further than this with beginners it is unnecessary to go.
As for Pythagoras, it is enough to give beginners one or
:wo of the many wellknown dissection figures.
Here T = W f V.
Fi. 88
Here the big square is cut up to form the two little
squares,
P = Q = R = s = P' =. Q'  R'  S',
M = M'.
The fact must be emphasized that in
:hese early stages any attempt at
formal proof is out of place. Never
:heless adequate reasons may be
xnmd, and should be provided, in
mpport of all statements made con
cerning these fundamental proposi
ions.
Here is a lesson on the centre of the circle as a centre
>f symmetry.
260
CRAFTSMANSHIP IN MATHEMATICS
85. The centre of the circle as a centre of symmetry.
Fold a circle on a diameter AB as an axis of symmetry,
prick through the two halves at C, open out and call the
corresponding points Q and C 2 , join C a and C 2 and join
each to the centre O, thus forming the isos. A C 1 OC 2 .
The arc of the sector AQOCg and the arc of the segment
ACjCg are the same.
Take a piece of celluloid (a piece of tracingpaper will
do, if you use it carefully), pin it down over the circle (fig.
90, i) by means of a pin thrust through it and through the
Fig. 90
centre O, and trace on the celluloid the radii OCj, OC 2 ,
the chord QCg, and the arc C 1 AC 2 \ now rotate the celluloid
round the pin (fig. 90, ii).
Since the traced sector and segment on the celluloid
preserve their shape and size while rotating, all lengths
and angles remain unchanged, and the arc C^ACg is seen
always to fit exactly on the circumference below it. We say
that the rotating sector and segment are symmetrical with
respect to the centre of the circle, because of this exact
fitting during the whole of a rotation. Thus the length of
the arc, the length of the chords, the angle between the
radii, all remain constant. We see all this plainly in fig. ii,
where the rotating sector and segment are shown in two
positions. Hence:
86. Equal chords in a circle are equidistant from
the centre. (L.)
ELEMENTARY GEOMETRY 261
87. Equal chords in a circle are subtended by equal
arcs. (L.)
88. Equal angles at the centre of a circle are sub
tended by equal chords and by equal arcs. (L.)
89. If two chords of a circle are equal, they cut
off equal segments. (L.)
90. All these things ( 8689) which apply to one
circle also apply to equal circles, since equal circles will fit
together exactly.
Early Deductive Treatment
Do not expect any rigorous logic from beginners. We
suggest a lesson easily within the comprehension of young
boys at the end of their First Year. Note: (1) the proofs
though of the simplest kind are enough to convince young
boys; (2) there is a logical grouping of the different kinds
of parallelograms; (3) the gradual extension of the properties,
as the variety within the species becomes more particularized,
is brought out. This gradual extension of properties should
always be borne in mind in the teaching of geometry.
Quadrilaterals as Parallelograms
91. A quadrilateral is a plane rectilineal figure
with four sides. (L.) There are different kinds of
quadrilaterals. We will begin
with the parallelogram. / / / / /
92. Draw a few parallel / / / / /
transversals across the parallel / / / / /
lines of your notebook. You / / / / /
see a number of foursided Fig. 91
figures with their opposite sides
parallel. These are parallelograms. (Gram means line.)
How can we define a parallelogram? First put it into
its class. ( 73.)
A parallelogram is a quadrilateral . . .
262
CRAFTSMANSHIP IN MATHEMATICS
What special property distinguishes a parallelogram? Its
opposite sides are parallel. Hence the definition:
A parallelogram is a quadrilateral with its opposite
sides parallel. (L.) Now let us discover the other properties
of a parallelogram.
93. Any side of a parallelogram may be regarded as a
transversal across two parallel lines. Let ABCD be a parallelo
gram, with the four angles
marked as shown.
Fig. 93
a + p = 2 rt. Zs,
p 4. y = 2 rt. Zs;
/. a f (3 = p + y.
/. a  y.
Similarly, p = 8.
(84)
(84)
Thus, the opposite angles of a parallelogram are equal.
Fig. 93
94. Join two opposite vertices by a line. Such a line is
called a diagonal. This diagonal (AC) is a new transversal
to both pairs of parallel lines.
Mark the two pairs of equal
alternate angles, a l9 2 , ft, ft,
and the pair of equal opposite
angles, y 1? y 2 . In the two Z.s
ABC, ADC, the diagonal forms
a side belonging to both, and the three angles of the one
are equal, respectively, to the three angles of the other.
Hence the two As are congruent.
.'. AB = CD and AD = BC.
Thus, the opposite sides of a parallelogram are
equal. (L.)
95. Draw the two diagonals, intersecting at E; and
consider the two triangles AEB, CED. In these As, the
three /.s of the one are equal to the three /.s of the other
( 82), and AB = CD ( 94). Hence the As are congruent.
/. AE = CE and BE = DE.
ELEMENTARY GEOMETRY
263
Thus, the diagonals of a
parallelogram bisect each
other. (L.) (Of course we
might have used the other As
AED and EEC.)
96. We may now collect
up the various properties of a
parallelogram:
Fig. 94
In any Parallelogram,
(1) The opposite sides are parallel, (known from the
(2) The opposite angles are equal.
(3) The opposite sides are equal.
definition)
(proved)
(proved)
(4) The diagonals bisect each other, (proved)
97. Imagine a 17 m to be jointed at the four angles, and
let it move from position A to position C. It remains a
Fig. 95
ZZ7m all the time, and therefore keeps all its properties. But
on its journey to C it passes through B, where the angles
are right angles. A parallelogram with right angles is called
a rectangle. A RECTANGLE is a rightangled parallelo
gram. (L.) Since a rectangle is a ZI7m, it has all the
properties of a I7m ( 96); and it has certain additional
properties:
98. (i) The angles of a rectangle are right angles.
(This follows from the definition.)
99. (ii) Draw the two diagonals of
the rectangle ABCD, and examine the
two As ABC and DCB (which, it will
be seen, partly overlap and have a
common base). The two sides AB, BC are equal to the
Fig. 96
264 CRAFTSMANSHIP IN MATHEMATICS
two sides DC, CB ( 96, 3); and the included angles ABC
and DCB are equal, both being right angles. Hence the
As are congruent. Therefore AC is equal to BD. Thus,
the diagonals of a rectangle are equal. (L.)
100. Suppose a rectangle gets shorter and shorter until
its length and breadth are equal. It remains a rectangle
Fig. 97
all the time and therefore keeps all its properties. When
the length exceeds the breadth, as in A and B, the rectangle
is called an oblong; when its length and breadth are equal,
it is called a square.
101. An OBLONG is a rectangle with its length
exceeding its breadth. (L.)
102. A SQUARE is a rectangle with all four sides
equal. (L.) Since a square is a rectangle, it has all the
properties of a rectangle ( 9799); and it has certain
additional properties:
103. (i) All four sides of a square are
equal. (This follows from the definition.)
104. (ii) Draw the two diagonals of the
square ABCD, intersecting in E, and examine
Fig. 9 8 the two As EEC and DEC.
BE = DE, ( 95, 97)
BC  DC, ( 103)
EC is common to both As;
/. ABEC = A DEC;
.'. ZBEC = /.DEC;
.*. each Z = J st. Z or 1 rt. Z.
Similarly we may show that each of the other two Zs at
E are rt. Zs. Hence, the diagonals of a square bisect
each other at rt. Zs. (L.)
105. We still require names for the two nonrectangular
parallelograms. The nonrectangular parallelogram with all
four sides equal is called a rhombus (fig. 99, i). The non
ELEMENTARY GEOMETRY
265
Fig. 99
rectangular parallelogram with only its opposite sides equal
is called a rhomboid (ii). The rhomboid is the parallelogram
we began with ( 9396). It
is the most general form of / / / /
parallelogram. If we made all i
its sides equal, or if we made
all its angles right angles, we should make it a particular
kind of parallelogram.
106. Just as an oblong may be reduced in length and
made a square, so a rhomboid may be reduced in length
and made a rhombus.
Just as a square has all the properties of an oblong and
certain additional properties, so a rhombus has all the pro
perties of a rhomboid and certain additional properties.
107. All four sides of a rhombus are equal. (This
follows from the definition.)
108. The diagonals of a rhombus are the perpen
dicular bisectors of each other. (L.) This property
may be discovered in this way. The word
rhombus really means a spinningtop. If
we stand it on an angle, it looks something
like a spinningtop. We see at once that
either diagonal is an axis of symmetry, B
and D being corresponding points about
the axis AC, and A and C being corresponding points about
the axis BD. Thus each diagonal is the perpendicular
bisector of the other.
But this also applies to a square. What is the difference
between a rhombus and a square?
Fig. 101
109. If we lengthen (or shorten) equally the two halves
of one of the diagonals, AB, of a square, we stretch out (or
266 CRAFTSMANSHIP IN MATHEMATICS
contract) the square into a rhombus. The diagonals of a
square are equal; those of a rhombus are unequal.
A rhombus differs from a square in these ways:
1. Its angles are not right angles.
2. Its diagonals are not equal.
A rhombus resembles a square in these:
1. The four sides are equal.
2. Each diagonal is the perpendicular bisector of the
other.
3. Each diagonal is an axis of symmetry.
110. We may classify the four kinds of parallelograms in
this way:
Parallelograms
Rectangles Non rectangles
Square Oblong Rhombus Rhomboid
(all sides eq.) (only opp. sides eq.) (all sides eq.) (only opp. sides eq.)
How easy it is to make up definitions from this scheme:
111. A SQUARE is a rectangular parallelogram
with all four sides equal ( 102).
112. An OBLONG is a rectangular parallelogram
with only its opposite sides equal ( 101).
113. A RHOMBUS is a non rectangular parallelo
gram with all four sides equal. (L.)
114. A RHOMBOID is a non rectangular parallelo
gram with only its opposite sides equal. (L.)
115. We might classify parallelograms according to their
axes of symmetry. A square has four axes of symmetry, viz.
two diagonals and two medians (a median is the line joining
the middle points of opposite sides); an oblong has two,
ELEMENTARY GEOMETRY 267
viz. the two medians; a rhombus has two, viz. the two
diagonals; a rhomboid has none. But in definitions we
do not usually refer to symmetry; symmetry is useful mainly
for discovering properties.
Remember that the two halves of a figure folded on an
axis of symmetry will fit together exactly. Remember, too,
that a figure can always be imagined to spin on an axis of
symmetry.
116. More about definitions. We have defined a
square as a rectangle with all four sides equal. There
fore,
(i) Since it is a rectangle, it has right angles and is a
parallelogram ( 97).
(ii) As it is a parallelogram, its opposite sides are
parallel ( 92). Thus, our definition of a square tells
us three things:
1. The four sides are equal,
2. The four angles are right angles,
3. The opposite sides are parallel.
But the definition tells us nothing at all about the diagonals.
Properties of the diagonals must be discovered either by
congruence ( 104) or by symmetry.
117. We might define a square as a quadrilateral with
four equal sides and four right angles. But all that this defini
tion tells us is that:
1. The four sides are equal,
2. The four angles are right angles.
It is quite a good definition, but it does not tell us that the
square is a parallelogram, and therefore it does not tell
us that the opposite sides are parallel. Hence this property
is one we should have to find out (perhaps by congruence) if
we used the new definition. Let us decide not to use it.
118. We can now classify the properties of parallelo
grams:
268
CRAFTSMANSHIP IN MATHEMATICS
Rhomboid.
Oblong.
Square.
Rhombus.
3
a
D
n
1. Opp. sides .
X
X
X
X
2. Opp. sides eq.
X
X
X
X
3. Opp. Zs. eq.
X
X
X
X
4. Diags. bisect each other.
X
X
X
X
5. All four Zs. rt. Zs.
X
X
6. Diags. eq.
X
X
7. All four sides, eq.
X
X
8. Diags. at rt. Zs.
X
X
Observe that the rhomboid is the most general of the parallelo
grams, and has fewest properties; and that the square is the
most special of the parallelograms, and has most properties.
No other parallelogram has all the properties of the square.
This lesson may usefully be followed up by the considera
tion of quadrilaterals that are not parallelograms.
Proportion and Similarity
A knowledge of proportion and similarity is so fruitful
throughout the whole range of the study of geometry that
the subject should be introduced at an early stage, though
naturally incommensurables are then ignored entirely. We
append a lesson suitable for the second year of the geometry
course. Note the little device for constructing a triangle
with sides simply commensurable. The proofs given are
rigorous enough at this early stage. The important thing is
to provide learners with a serviceable weapon rough and
unpolished, for the moment, it is true; but that is of no
consequence.
(The nature of a ratio, of crossmultiplication, &c., has
already been referred to in the chapters on arithmetic and
algebra, but the three subjects should be brought into line
when a principle common to them all is under consideration.)
ELEMENTARY GEOMETRY 269
119. Take a piece of paper ruled in J" squares, and on
it draw this triangle: the base AB of the A is to be on one
of the ruled horizontal lines 4" or 5" down the paper, and
the vertex in a parallel line 3"
above, i.e. in the twelfth parallel
line above. Fix the point A
towards the lefthand end of the
line selected for the base, and with
a radius of 36" draw a circle to
cut the top line in C. With C Fig. 102
as centre, and with a radius of
45", draw a circle to cut the base line in B. Join AB (its
length does not matter) and so complete the A ABC.
Since AC = 36", it can be divided into twelve equal
parts of 3" each, and each division will fall on one of the
ruled horizontal lines. Since CB ^  4i", it can be divided
into twelve parts of f " each, and again each division will fall
on one of the ruled horizontal lines. But the ruled lines are
all parallel to each other. We therefore seem to have the
following result:
120. If the two sides of a triangle are divided into
the same number of equal parts, and the corresponding
points of division in the two sides are joined, all the
joining lines are parallel to the base. (L.)
It has been found that this result is always true, no matter
how it is tested. But the real proof is too difficult for you
to understand at present. The following particular case is
often useful:
121. If two sides of a triangle are bisected, the line
joining the points of division is parallel to the base.
(L.)
122. We may now learn that if
a line is drawn parallel to one side
of a A , it cuts the other side pro
portionally. Consider, for instance,
the fifth parallel DE, from the top
in the figure to 119. CD is  x \ of Flg> I03
270 CRAFTSMANSHIP IN MATHEMATICS
CA, and DA is & of CA; CE is & of CB, and EB is / 2  of
CB.
CD 5 , CE 5
nnn
" DA~7 EB~7'
CD ^ CE
" DA " EB'
So with any other parallel. Or a part of a side may be compared
with the whole. For instance,
CD ^ CE
CA ~ CB'
for each is equal to the fraction  L ! V
We can imagine the A CDE to be a small A fitting over
the top of the larger A CAB (fig. 104, i), and CD being made
to slide down CA so that the small A CDE occupies the
\L _\r
Fig. 104
position C'D'E', D taking the place of A (fig. 104, ii). Just
as the three /.s of the small A are respectively equal to the
three Ls of the large A in fig. 104, i (see 78), so they
must be in the second, since corresponding angles are equal.
Hence C'E' is  CB, and therefore C'E' cuts the two sides
D'C and D'B proportionally; and just as C'D' is {\ of CD',
so D'E' must be ^ of D'B,
D'C' D'E' D'C' D'E'
or = . or = .
C'C E'B' D'C D'B
Similarly by making the little A slide down to the other
corner (fig. 104, i to iii), so that E takes the place of B, C"D"
is  to CA, and therefore
E // C " E"D" E"C" E"D"
 . Q
C"C D"A' E"C E"A'
Hence:
ELEMENTARY GEOMETRY 271
123. If in a triangle a line is drawn parallel to
any side, it cuts the other sides proportionally. (L.)
(This is always true, but the real proof is too difficult for
you to understand at present.)
124. We might detach the small ACDE from the large
one CAB, and place the two side by side. They look alike
They are alike. They are similar. c
Although the sides of the two As
differ so much in length, the three
Z.s of the one are respectively equal to A / \ B
the three /.s of the other. (Why?) b lg . I05
In other words, the two similar As are
equiangular. And we know already that the corresponding
sides are proportional. This we should expect in similar
figures of any kind. In a photograph of yourself, for
instance, you would expect the " proportions " of your body
to be accurately preserved. If the ratio of the lengths of
your outstretched forearm and upper arm is f , you would
expect that ratio to be preserved in the photograph (or the
photographer would probably hear about it!).
125. If, then, ABC and DEF are two similar As, the
corresponding sides are proportional. But note that we
may express the ratios in two
different ways: (1) two sides of
one A as a ratio equal to the
ratio of the corr. two sides of the
other A ; (2) one side of one A p lfg . I0 6
and the corr. side of the other
A as a ratio equal to the ratio of any second side of the
first A and the corr. side of the second A . Consider, for
instance, the two sides AB, BC in the A ABC, and the two
corr. sides DE, EF in the A DEF. We may say,
AB DE AB BC
BC'EF' r DE = EF
The two proportional statements are really the same thing
272 CRAFTSMANSHIP IN MATHEMATICS
since we obtain the same product from the crossmulti
plication of either:
AB . EF = BC . DE. (The full stop is used instead of X .)
It is often an advantage to interchange one form for the
other; really we interchange the second term of the first
ratio and the first term of the second.
We have learnt that:
126. SIMILAR TRIANGLES are equiangular, and
their corresponding sides are proportional. (L.)
127. When expressing ratios between two sides of each
of two similar As, be careful to select corresponding
sides, i.e. sides taken in the same order round corre
sponding angles. In these two pairs of As, the corre
sponding Zs are marked with the same Greek letters. From
the first pair we may equate ratios thus, six equations in all:
AB = DE. AB _ DE. AC _ OF. AB _ BC. AB _ AC. AC _ BC
BC EF ; AC DF ; CB FE J DE EF ; DE DF ; DF EF*
From the second pair, we may do exactly the same thing:
=Q5. MP^QS. MN^NP. M^MP. MP__PN
QS ; PN SR ; QR RS ; QR QS ; QS SR*
Yet there appears to be a difference. That is because in
the second pair the As are right and lefthanded. If you
have any doubt, turn one of the pair over, through 180,
as you would turn over a page of a book. Then the pair
will look alike. But if you mark the corresponding angles
correctly, you ought to have no difficulty.
ELEMENTARY GEOMETRY
273
The sides of the triangles may usefully be named by
means of single small letters; then the writing of the ratios
is simplified; e.g.  instead of .. The small letter selected
a BC
for a side is always the same as the capital letter naming the
opposite angle.
128. Sometimes each of a pair of similar As is similarly
divided by a perpendicular from a vertex to the opposite
side. The resulting pair of A
As in the one case are evi
dently similar, respectively,
to the resulting pair in the
other case, for they are
. ~~ u ~ "
equiangular, i.e. As ABG iv ]g . 10 s
and DEH are similar, and
As ACG and DFH arc similar (check, by sum of angles).
We may reason in this way:
In the As ABG, DEH, since
AG_ = AB
DH DE
4, 4,
AB
and in the As ABC, DEF, since =
I
. AG _ BC
" DH EF*
An exchange of ratios may often usefully be made in this
way. From this particular exchange we learn that the altitudes
AG, DH are proportional to the bases BC, EF. Hence:
129. When similar triangles are divided by perpen
diculars drawn from corresponding vertices to opposite
sides, an exchange of ratios may often be usefully
made. (L.)
130. In similar triangles, the altitudes are pro
portional to the bases. (L.)
131. We know that when two ratios are equated to form
a proportion, they may be cleared of fractions by cross
(E291) 19
274
CRAFTSMANSHIP IN MATHEMATICS
multiplication. For instance, in the two similar As ABC,
DEF, we know that
= ; (an equation consisting of 2 ratios)
BC EF
/. AB . EF = DE . BC. (an equation consisting of 2 products)
What does this mean? AB, BC, DE, EF, all represent lines
of a particular length; a length multiplied by a length gives
an area. Thus each of the two products AB . EF and DE . BC
represents a rectangle.
c e
625
Fig. 109
Note that we begin with ratios, i.e. with quotients,
representing a length divided by a length. After cross
multiplying, we have products representing areas, or a
length multiplied by a length. (The measured lengths are
shown to scale. Check the numerical ratios and the products.
625 3'75
For instance, are and equal? and are 10 X 375
and 6X6 25 equal?) 10 6
132. By cross multiplication, two equated ratios
of lengths give two equated rectangular areas.
(" Equated " means expressed as an equation.)
133. The last result is useful in all
sorts of ways. For instance, AB and
CD are two chords of a circle, inter
secting at O. Join AC and DB, and
we have two similar As, the As being
equiangular (angles in the same segment;
see 126). Taking ratios (see 127)
we have,
Fig. no
OA
oc :
rect. OA . OB
op
OB ;
rect. OC . OD.
ELEMENTARY GEOMETRY
275
Hence, if two chords intersect in a circle, the rectangle
contained by the two segments of the one is equal to
the rectangle contained by the two segments of the
other. (L.) (The term " segment " here applies to the
parts of the chords.)
Fig. in
The proportional division of lines and the construction of
similar figures should follow at a slightly later stage. Be sure
that the boys master the principle exemplified in these three
figures: for the construction of the similar pentagon the
position of. the point O is quite immaterial.
The centre of similarity problems are readily followed by
those on centre of similitude. Insist on the point that any
two circles may be regarded as similar figures, since, like
rectilineal similar figures, they may be looked upon as the
same figure drawn to different scales.
Circles and Polygons
The ordinary properties of the circle give little trouble
angles in a segment, the cyclic quadrilateral, tangents,
alternate segment property, circles in contact, and inter
secting circles. Do not forget to group properties around a
276
CRAFTSMANSHIP IN MATHEMATICS
common principle; e.g. (i) the tangent to a circle, (ii) the
external common tangent to two circles, (iii) the transverse
common tangent to two circles, should be taken in that order,
and be made to follow on the key proposition that the angle
in a semicircle is a right angle. All these propositions on the
circle being quite simple, formal proofs should now be
consistently exacted.
Regular polygons, too, need give little trouble. Their
angle properties are interesting, easy to understand, and
always appeal to a boy. The pentagon excepted (see the
next section), they are not much
wanted. The hexagon and octagon
involve the simplest geometry, easy
work for beginners. The decagon is
easily constructed from the pentagon,
and the dodecagon from the hexagon.
The heptagon and nonagon are hardly
ever used; the latter is easily constructed
from its angle properties; the former
is riot, inasmuch as its angle properties
involve fractions of a degree and hence
some sort of approximation method is
required for its construction. The
best is probably the following, especially as it is common to
all polygons.
On one side of a straight line draw a semicircle and on
the other side an equilateral triangle. If the line be divided
into x equal parts, and lines be drawn from the apex of the
triangle through the points of division, to meet the semi
circle, the semicircle is divided into the same number of
parts as the line. This is not susceptible of proof, simply
because it is not mathematically true, but the approximation
is so near that the most careful measurement usually fails
to detect an error. Architects generally use it. Evidently
by drawing radii from the points of division of the semicircle,
we divide 180 into x equal parts.
The conversion of polygons (regular and irregular) into
ELEMENTARY GEOMETRY 277
triangles, triangles into rectangles, and rectangles into
squares, which is often wanted, is simple straightforward
work, though some little practice in manipulating the figures
is necessary. To the beginner, a polygon with one or more
reentrant angles is puzzling.
Golden Section and the Pentagon
We append the following lesson as an example of linking
up different Euclidean propositions (II, 11; IV, 10, 11) and
of utilizing algebra in solving geometrical problems.
134. To divide a line into two parts so that the
rectangle contained by the whole and one part is equal
to the square on the other part.
This is sometimes stated:
To divide a line in medial section.
or, To divide a line in extreme and mean ratio.
or, To divide a line in golden section.
The problem is very easy to do and to understand if we
can solve easy quadratic equations. It is the kind of problem
in which algebra can help us
much, A , x L(2z) > B
Let AB be the line to Fig II3
be divided, and let it be, say,
2" long. Suppose the point of division is P.
Let AP be x inches long; then PB = (2 x) inches long.
The line has to be divided so that AB . BP = AP 2 ,
i.e. 2(2 x) = x 2 .
We must now solve this equation, and find the value of x
X * = 2(2  *);
.'. x 2 + 2x = 4.
/. x 2 + 2x + I = 5.
.*. x + I = + V5. (We may neglect the minus sign.)
/. X = V5  1,
278
CRAFTSMANSHIP IN MATHEMATICS
i.e. AP = (\/5 1) inches. Can we measure off this length
and so find P? Yes, by the theorem of Pythagoras. We do
it in this way:
Erect a JL BC at B, 1" long, and join CA.
AC 2  (AB 2 + BC 2 ) = (2 2 + I 2 ) = 5;
/. AC = V5,
i.e. AC is V5 inches long (fig. 113a, i).
Fig. 1130
But we require a line (\/5 1) inches long.
Since CB I", with centre C and radius CB, cut CA in
D (fig. 1130, ii); CD = I".
Thus AD  (V5  1) inches.
But we require a part of AB equal to (\/5 1) inches.
Hence, with centre A and radius AD, cut AB in P;
AP = (<v/5 1) inches.
Thus P is the point required.
The length of PB is evidently 2 (<v/5 1) in., i.e.
(3 V5) inches.
If we have done " surds " in algebra, we can show
that the result is correct: AB . BP has to be equal to
AP 2 . Now AB . BP  2(3  ^5) = 6  2 V5; and AP 2 
(V5  I) 2 = 6  2v/5, as before.
Here is Euclid's figure. He does not
cut off a piece from CA; he makes CD
equal to CA, so that CD = <v/5, and BD
== \/5 ! Then he makes BP equal to
BD = \/5 1, so that P is found as before,
except that PB is now the longer instead
Fig. ii 4 of the shorter section. The shaded parts
of the figure show the rectangle AB . AP
2(3 V 5 ); and the square on PB, (^/5 I) 2 .
ELEMENTARY GEOMETRY
279
Now examine a regular pentagon and its 5 contained dia
gonals. Give the boys a few hints (such as the following)
and then leave them to construct the pentagon themselves.
(1) Angles. 04 = a 2 = a 3 ; hence it is clear that the
15 angles at the 5 vertices of the pen
tagon are all equal, and that each =
36.
(2) Lines. Each diagonal is divided
by 2 others into 3 parts. Is there
any relation between the parts? e.g.
does CF bear any relation to FA or to
the side CD?
Draw the triangle ACD and the
line FD separately, and write in all
the angles. Evidently AFD and DFC are isosceles triangles.
. . AF = FD, CD = FD, / . AF = CD. Hence if we put
a circle round the triangle AFD, CD is a tangent (relation
Fig. 115
Fig. 116
= ?); also CA is a secant; / . CF . CA = CD 2 ; / . CF . CA
= FA 2 , i.e. CA is divided at F in golden section.
To construct a pentagon, therefore, we begin by drawing
any line AC, and dividing it in golden section in F. With
A as centre and AC as radius, we draw a circle (not shown)
and draw in it the chord CD equal to FA, and then join
AD. This gives us the triangle ACD, round which we
circumscribe a circle and so obtain part of fig. 115; to obtain
the points B and E we bisect the angles ADC, ACD.
280 CRAFTSMANSHIP IN MATHEMATICS
Teach the boys one or two special ways of drawing the
pentagon; e.g. let them tie into a simple knot a strip of paper
of uniform width. It is a useful exercise to make them prove
that the figure produced really is a pentagon.
The Principle of Continuity
This is an ambiguous term, for in each of several branches
of knowledge it is given a special significance. Even in the
single subject mathematics, it is used in different senses.
One standard textbook of geometry states: " The principle
of continuity, the vital principle of modern geometry, asserts
that if from the nature of a particular problem we should
expect a certain number of solutions, and if in any particular
case we find this number of solutions, then there will be
the same number of solutions in all cases, although some of
the solutions may be imaginary. For instance, a straight
line can be drawn to cut a circle in two points; hence we
state that every straight line will cut a circle in two points,
although these may be imaginary or may coincide. Similarly
we may say that two tangents may be drawn from any point
to a circle, but they may be imaginary or coincident/' *
But in geometry the term " continuity " has come to be
used more loosely than that. It is used to indicate generality,
a generalizing of some fundamental principle, or the grouping
of a number of allied instances around some central principle.
We give a few instances of different kinds, from which the
* Lachlan, Modern Pure Geometry.
ELEMENTARY GEOMETRY
281
reader will see more clearly what is meant. As regards the
teaching of geometry, the principle is one of the very greatest
importance.
1. The particularizing of a general figure and the extension
of properties. We have already given an instance of this in
the lesson on parallelograms.
2. Varying the figure to include different cases. These
three figures tell their own story. If the parts of such figures
Fig. 118
are similarly named, as a rule exactly the same words apply
in all cases to the proof. What difference there is is generally
a difference of mere sign.
3. Generalizing a term to include its natural extensions,
Fig. 119
e.g. a chord as a secant, and a secant as a tangent. From the
case of intersection O inside the circle, we pass to the case of
intersection outside the circle, and then from the two secants to
a secant and a tangent. The three cases may first be separately
282
CRAFTSMANSHIP IN MATHEMATICS
taken and then generalized. If the lettering is consistent,
the arguments are identical, though for the tangentsecant
case we should generally argue rather differently. In all
three cases we have two similar triangles, OAC and OBD,
and OA/OC = OD/OB.
Another general chordsecanttangent property is seen in
the following four figures, showing the measure of an angle
Fig. 1 20
inscribed in a circle by reference to the intercepted arcs;
again the argument may be made perfectly general.
4. The extensions of Pythagoras form another series. It
is the general custom nowadays to give the boys Pythagoras
towards the end of their first year, to serve as a useful working
tool; to give a formal proof during the second year, and to
take the extensions (Euclid II, 12, 13) a few months later
still; most boys are then familiar with the results in the
following form.
AB 2 = AC 2 1 CB 2 exactly.
AB 2 =
AC 8 + CB 2 + 2BC.CP
AC 8 + CB  2BC.CP
where CP is the
projection of
CA on BC.
(L.)
ELEMENTARY GEOMETRY
283
But figures to illustrate the extensions are less often pro
vided. Here is a suggestion:
Fig. 121
Fig. (i) illustrates Pythagoras, In (ii) compare the squares
on the new sides BY, YA with the squares on the old sides
BC, CA. In (iii) compare the squares on the new sides
BZ, ZA with the squares on the old sides BC, CA. The
dissections are interesting, though they tend to puzzle slower
boys.
5. Summing the exterior angles of a polygon: " walking
the polygon ". First consider an ordinary convex polygon.
Mark in the angles systematically: " always turn to the
284 CRAFTSMANSHIP IN MATHEMATICS
left "; the angle to be worked is that between the old
direction and the new.
SUM op Ls. 2ir
Fig. 1220
Secondly, a polygon with one reentrant angle:
SuMOfU4ir
Fig. 1226
Thirdly, a cross polygon; also with one reentrant angle:
c
D SUM op Ls
Fig. I22C
The point in this example is to see how exactly the same
principle is followed out: always turn to the left, always
measure the exterior angle between the old direction (pro
duced) and the new. The result must always be a multiple
Of 27T.
6. Euclid, Book IL Given an algebraic basis, suitable
ELEMENTARY GEOMETRY 285
figures, and a rational grouping, props. 4 and 7, 5 and 6,
9 and 10, can be taught in a single lesson. Never make the
boys go through the Euclidean jargon; life is not long enough.
7. The Sections of a Cone (for more advanced pupils).
Let a plane perpendicular to the plane of the paper rotate
round the point P, first cutting the cone ABC parallel to
the base; then obliquely to cut the slant surface; then more
obliquely, parallel to AC and cutting the base; then per
pendicularly to the base and cutting the base. Since the
motion may be regarded as continuous, we should expect
no sudden changes in the properties of the curves made by
the rotating plane as it cuts the cone. Why
should there be? The boys' knowledge of *
geometry ought by this time to make them P / \
revolt against the idea of any fundamental
differences in the properties of the curves.
The curves may all be described as conies
possessing certain common properties. In
\
particular positions the curves have certain Fig. 123
additional and special properties, but the
common properties will remain. Let the boys understand
that for convenience we study the curves separately first,
and collectively later. But make them see at the outset
that the circle is just a particular case of an ellipse, just
as the ellipse is a case of the more general conic. The
elliptic orbit of the earth, for example, is so very nearly a
circle that a correct figure drawn on paper is virtually in
distinguishable from a circle. Astronomical figures are
often purposely exaggerated.
8. The Polyhedra (see Chapter XXXVIII). These form
an even better illustration of the principle of continuity than
those already cited.
The principle applies, in fact, to the whole range of geo
metry. To deal with a proposition as an unrelated unit is,
generally speaking, to offend almost every canon of geometrical
teaching.
286
CRAFTSMANSHIP IN MATHEMATICS
The Principle of Duality
This is best exemplified by a few wellknown pairs of
theorems:
1. If the sides of a triangle
are equal, the opposite angles are
equal.
2. If two triangles have two
sides and the included angle
respectively equal, the triangles
are congruent.
3. If a quadrilateral be in
scribed in a circle, the sum of
one pair of opposite angles is
equal to the sum of the other pair.
4. If a hexagon be inscribed in
a circle, the three points of in
tersection of pairs of opposite
sides are collinear.
1. If two angles of a triangle
are equal, the opposite sides are
equal.
2. If two triangles have two
angles and the included side
respectively equal, the triangles
are congruent.
3. If a quadrilateral be circum
scribed about a circle, the sum of
one pair of opposite sides is equal
to the sum of the other pair.
4. If a hexagon be circum
scribed about a circle, the three
diagonal lines connecting opposite
angles are concurrent.
Such pairs of propositions are said be dual or reciprocal.
There is, in short, a remarkable analogy between de
scriptive propositions concerning figures regarded as as
semblages of points and those concerning corresponding
figures regarded as assemblages of straight lines. Any two
figures of which the points of one correspond to the lines
of the other are said to be reciprocal figures. When a pro
position has been proved for any figure, a corresponding
proposition for the reciprocal figure may be enunciated by
merely interchanging the terms point and line; locus and
envelope*, point of intersection of two lines and line of inter
section through two points] &c. The truth of the reciprocal or
dual proposition may usually be inferred from what is called
" the principle of duality ".
The teacher should always be on the lookout for examples
of this principle which gives boys so much insight into geo
metry. Numerous examples of concurrency and collinearity
will occur to him at once. The principle is especially useful
ELEMENTARY GEOMETRY 287
in the treatment of more advanced work, for instance in the
theory of perspective and in the theory of the complete
quadrilateral (tetrastigms and tetragrams).
CHAPTER XXI
Solid Geometry
Preliminary Work
First notions of solid geometry will have been given in
the Preparatory School. Even in the Kindergarten School
the children are made acquainted with the shapes of common
geometrical figures and solids. Lower Form arithmetic is
closely linked up with practical mensuration, and quite young
boys are made familiar with the methods of measuring up
rectangular surfaces and solids. The practical mensuration
associated with early measurements in physics forms another
introduction to solid geometry. First notions of projection
are given in early geography lessons; very young boys soon
acquire facility in building up vertical crosssections from
contoured ordnance maps, and when projection is first formally
taken up in the mathematical lessons, say in the Pythagoras
extensions or in early trigonometry, the main idea is already
familiar. All the way up the school, threedimensional
geometry in some form should be made to serve as a hand
maid to the plane geometry. Indeed, first notions of the
geometry of the sphere are required at a very early stage in
the teaching of geography, and if these notions are to be
properly implanted the mathematical Staff should make
themselves responsible, for not all geography teachers are
mathematicians.
Only a minority of boys acquire readiness in reading
geometrical figures of three dimensions. With the majority,
288 CRAFTSMANSHIP IN MATHEMATICS
the training of the geometrical imagination is a slow business.
For the clear visualization of the correct spatial relations in
an elaborate threedimensional figure, or for that matter even
in a simple one, models of some kind are, in the earlier stages,
essential.
Supplies of useful little wooden models of the geometrical
solids are often found in the physics laboratory, though why
physics teachers so frequently relieve their mathematical
colleagues of this particular work I have never been able to
discover. If models in wood are not available, models may
be readily cut from good yellow bar soap; the material is
cleaner to handle than raw potato or clay or plasticine. By
means of a roughlycut model, the correct shape of a trans
verse section of a geometrical solid can be realized at once.
Personally I prefer models made from " nets " of cartridge
paper or thin cardboard; these are easy to make and are
permanent, but the making consumes a good deal of time.
Useful skeleton models are readily made from pieces of
long knitting needles, sharpened at each end and thrust into
small connecting corks. Two slabs of cork to represent
the Horizontal and Vertical Planes, tacked to a pair of hinged
boards, and a few pointed knitting needles, make excellent
provision for the first lessons on orthographic projection.
The natural sections of an orange, or the cut sections of
a wellshaped apple, are useful when teaching the geometry
of the sphere.
The small varnished wooden models of the cylinder,
sphere, and cone, of the same diameter and height, are useful
for showing, by displacement of water in a measuring jar,
that the volumes are 3:2:1.
A slated sphere, mounted, should be part of the equipment
of all mathematical teachers.
Even such a simple device as two intersecting sheets of
paper, each sheet being slit halfway across, to show the
intersection of two planes at any angle, is often useful.
But of course all these props should gradually be with
drawn, and the eye made to depend on twodimensional
SOLID GEOMETRY 289
drawings. Still, it is always an advantage, even for the trained
mathematician, to put a few shading lines into such drawings.
They help the eye greatly.
Stereographic photographs, or even handmade stereo
grams, are also a great aid in teaching solid geometry. These
are easily provided, and stereoscopes are cheap. Mr. E. M.
Langley used them with great effect as far back as the nineties.
Do not forget that even for plane geometry models may
be useful. The pantograph is particularly useful when teaching
similarity (see Carson and Smith's Geometry). When teaching
loci, encourage the boys to make wooden or cardboard
" linkages " to represent engineering motions and astro
nomical movements. The loci are then given a reality.
The boys should also be encouraged to make " nets "
of the commoner geometrical solids, in cartridge paper or
cardboard. Boys of 11 or 12 learn to make these readily,
and at that age time can be spared. I have known boys of
10 make almost perfect paper models of the five regular
polyhedra.
In naming triangular pyramids, name the vertex first,
then the three corners of the base, thus, A. BCD. Note that
any corner of such a pyramid may be regarded as a vertex,
the other three being the corners of the base (just as any
corner of a A may be regarded as a vertex, and the other
two corners as the ends of the base).
A problem like the following is better understood if a
prism is actually cut up, perhaps a wooden one made in the
carpenter's shop; or one may be cut neatly from a bar of
soap. It is easier to cut up the latter with a thin knife
than to cut up the former with a saw.
A suitable model for showing that a prism is equal to
three times the volume of a pyramid on the same base and of
the same height is a little troublesome to make. Inasmuch as
it is particularly useful in the demonstration of that important
principle, we give a few hints for constructing it.
Fig. 124, i, represents the complete triangular prism,
with bases ABC, DBF. From it, cut the pyramid E.ABC
(E291) 20
290
CRAFTSMANSHIP IN MATHEMATICS
by holding the knife (or saw) at E, and cutting through to
AC. iii shows the pyramid cut off.
Now we have to cut the remaining piece (ii) into two
other pyramids. Cut from it the pyramid C.DEF. To do
this, hold the knife again at E, and cut down to DC. v shows
the pyramid cut off, its new face being shaded, iv shows
the part left. It is a curiouslooking wedgeshaped pyramid.
We will name it E.DAC.
We may show (since pyramids on equal bases and of the
same vertical height have the same volume) that the three pyra
mids (iii, iv, v) are equal in volume. The bases of iii and v,
E.ABC and C.DEF, are equal, since they are the bases of the
prism; and the heights are equal, for FC EB, and these are
two of the long edges of the prism. Again, if we name iv and
v E.ACD and E.FDC, we see that the bases are equal, for
they are the halves of DACF, one of the faces of the prism;
and their vertical heights are equal, since the two have a
common vertex E. Hence the volumes of all three pyramids
are equal.
It is a simple matter to make the " nets " (fig. 125) of
the three pyramids, and fold them up to make models. The
models may then be placed together to form the prism. Each
net will, of course, consist of four triangles, the sides of all
of which will be edges, or diagonals of the faces, of the prism.
The formal mensuration of geometry of the pyramid, then
of the cone, then of the cylinder, is interesting and valuable,
SOLID GEOMETRY
291
Fig. 125
and I do not find that it gives teachers much trouble, always
provided that the necessary preliminary work from Euclid
XI on lines and planes has been done well.
The calculation of the areas of the surfaces of solids is
also simple, including even the surface of the sphere, provided
that suitable figures are drawn.
Euclid XI
All the essential propositions from Euclid, Book XI, are
now included in the leading schoolbooks on geometry.
Most boys find the reasoning easy enough, but many have
great difficulty in understanding the figures, unless models
are available to help visualization.
It will suffice to touch upon Euclid XI, 4 and 6.
XI, 4. If a straight line is perpendicular to each of two
intersecting straight lines at their point of intersection, it is
perpendicular to the plane
containing them.
The first of the follow
ing figures is Euclid's
own, and to most boys
it is incomprehensible.
The second is that found Fig I26
in many modern text
books. This is a case where a model is certainly desirable.
Failing that, two figures should be drawn, from which the
292
CRAFTSMANSHIP IN MATHEMATICS
different planes may easily be picked out. The following
figures are suitable: in the first, the horizontal plane and the
vertical planes are easily seen; in the second, the two oblique
planes. If such figures are steadily gazed at, with one eye,
Fig. 127
through a very small hole in, say, a piece of cardboard,
they quickly assume an appearance of three dimensions.
XI, 6. If two straight lines are perpendicular to the same
plane, they are parallel.
The first figure is Euclid's (again a poor thing); the
second is that commonly found in school textbooks. In
Fig. 128
this case, again, the planes want sorting out, to help visual
ization. Figure 129 is more suitable, with the horizontal
plane shaded. The two perpendiculars AB, CD are shown
by rather thicker lines. The two congruent triangles FDA,
EDA in the oblique plane AFE are easily picked out; so
are the two BDE, BDF in the horizontal plane. But it is so
difficult to draw a figure that will show, to a beginner's eye,
SOLID GEOMETRY 293
the two congruent triangles ABE, ABF in their separate
vertical planes, that either a wire model or a pair of stereo
grams are certainly desirable.
Fig. 129
It is unfortunate that so many boys experience difficulty
in visualizing threedimensional figures. But the fact has
to be allowed for, and provision made accordingly.
Do not press too far the argument that such aids as models
should be withdrawn in order that the boys' imagination may
be given opportunity to develop. The boys' developed
imagination will be a poor thing if it has to be nurtured on
the teacher's badlydrawn figures.
CHAPTER XXII
Orthographic Projection
Elementary Work
Below we reproduce subjectmatter suitable for two or
three preliminary lessons on orthographic projection to the
Middle Forms. Time can seldom be found for much ruler
and compass work, but freehand drawings, rapidly executed
294
CRAFTSMANSHIP IN MATHEMATICS
in association with the teacher's own blackboard demon
strations, may be made to serve a useful purpose in laying
the foundations of the subject. Higher up the school, if time
permits, more advanced work should be taken. It helps the
ordinary geometry, plane and solid, greatly.
In preparing drawings for builders, architects make plans
and elevations of buildings to be erected. A plan of a
thing is an outline on a horizontal plane; an elevation is an
outline on a vertical plane.
Push the table up against the wall. On the table place
a rectangular block with two faces parallel to the wall. Chalk
on the table an outline of
the base of the block, and
thus make a plan of the
block. Now push the block
against the wall, and chalk
an outline on the wall.
This is an elevation of the
block; the elevation a'b'c'd'
in the figure is a projec
tion of the face abed of the
block.
It is a very simple kind
of projection, because most of the work to be done depends
on the drawing of perpendiculars and parallels. The
projectors and other working lines are nearly all perpen
diculars and parallels. A word implying perpendiculars
and parallels is " orthographic ", and the projection is some
times called orthographic projection.
As it is not very convenient to draw on the wall, we
sometimes use two boards hinged at right angles. The next
figure shows such a pair, first of all hinged in position, then
unhinged and the vertical plane turned back into the hori
zontal. The figure shows two plans and elevations of the hut
in fig. 136. The first plan shows the long sides of the hut
parallel to the vertical plane and the elevation a side eleva
Fig. 130
ORTHOGRAPHIC PROJECTION
295
tion. The second plan shows the long sides perpendicular
to the vertical plane, and the elevation an end elevation.
The term front elevation is also sometimes used. An ele
vation is often spoken of as a view.
An architect would not draw two plans of one building,
but he would always draw two or more elevations, in order
Fig. 131
to make the builder understand exactly what the building
was to be like.
Note that all the projectors (shown as broken lines) are
perpendiculars and parallels.
The two boards are shown hinged merely to help you to
understand how the plan and elevation are related, but plans
and elevations are commonly drawn as in fig. ii. An architect
would not trouble to draw the outlines of the two boards.
He just rules a line across the paper (marking it XY some
times), draws the plan below, and the elevation above. More
frequently than not, he uses separate pieces of paper, but
he always remembers how the separate drawings are related.
The XY line is sometimes called the ground line: it is the
line of contact of the vertical plane with the ground. It is
usual to keep the plan a little distance away from this line,
but to let all elevations stand on the line.
Although your plans and elevations will always be drawn
on the flat, you will sometimes find it useful to fold your
paper at right angles on the XY line, and to place the object,
if small enough, in position on the horizontal plane. You
can then see more plainly what the elevation on the vertical
296
CRAFTSMANSHIP IN MATHEMATICS
plane will be. For instance, in the figure at the beginning
of this section, suppose a pencil is placed in contact with
the edge ad of the block, and the block removed. You can
see at once that the plan of the pencil is bc y and that the
elevation is a'd'.
Remember that plans and elevations of any object are
obtained by drawing perpendicular projectors to the
Horizontal Plane (H.P.) and Vertical Plane (V.P.). The
feet of these projectors are then joined in such a way that
the lines correspond to the edges of the object itself.
Here are some examples of plans and elevations. Copy
them full size. Then fold your paper on the ground line
and turn the V.P. into position.
1. Plans and elevations of a line 3" long. Hold a piece
of wire or a short pencil in position, so that, looked at from
x j
567
Fig. 132
above, it covers the plan, and looked at from the front, it
covers the elevation. Then, in each case, try to describe
the position of the pencil with reference to the two planes,
checking your descriptions by the correct descriptions below.
i.
2
3.
4.
5.
6.
7.
To the H.P. the line is
II
JL
II
60

45
60
To the V.P. the line is
II
II
J_
II
45
45
30
2. Plans and elevations of a rectangular sheet of paper,
3" X 2":
ORTHOGRAPHIC PROJECTION
297
The Plane of
the Paper is
The Long
Edges are
The Plane of the
Paper is
The Long Edges are
1.
 to V.P.
 to H.P.
5.
 to V.P.
30 to H.P.
2,
II V P
JL H.P.
II H.P.
60 V.P.
3.
II ,, H.P.
II V.P.
7.
45 both planes
 both planes
4.
II H.P.
JL V.P.
8.
L ,, both planes
45 both planes
3. Plans and elevations of a square prism:
Positions as follows:
(1) Standing on base, two sides  to V.P.
(2) Lying on a side, all sides JL to V.P.
(3) Lying on a side, bases J to both planes.
(4) Standing on a base, one diag. of base J to V.P.
(5) Lying on a side, two sides 30 with V.P.
(6) Same as No. 1, with a section AB to V.P.
(7) Same as No. 5, with a section CD JL to H.P.
298
CRAFTSMANSHIP IN MATHEMATICS
It is sometimes necessary to know the shapes of sections
of solids. Two are shown above. The cut surfaces are in
dicated by cross hatching. When in doubt about such a
shape, make a rough model and cut through it.
The positions of objects with respect to the two planes
may be described in more than one way. In the first of the
last series, for instance, we might have said two sides JL to
V.P. In the third, we might have said two sides  to the
V.P. and two  to H.P. But the description must always be
sufficient to fix the object in a particular position.
4. Plans and elevations of other solids:
Fig. 135
Position of solids:
(1) Hexagonal prism standing on base, two sides  to V.P.
(2) Hexagonal prism lying on a side, long edges 30 to V.P.
ORTHOGRAPHIC PROJECTION 299
(3) Hexagonal pyramid standing on base, two edges of
base  to V.P.
(4) Cylinder standing on base, with section AB L to H.P.
(5) Cylinder lying down, axis 30 to V.P.
(6) Cone standing on base, with section CD L to V.P.
Sections which are cut obliquely to one of the two pro
jection planes may give a little trouble, especially if their
shapes cannot be first imagined. The shape of an oblique
section through a cylinder may be shown by half filling a
round bottle with water and holding the bottle obliquely;
the watersurface gives the shape of the section an ellipse.
So with a square bottle, or a conical flask. Or you may push
the solid obliquely into the ground, down to the level of the
section line. The shape of the mouth of the hole is the shape
of the section.
More Advanced Work
Here are types of problems suitable for more advanced
pupils:
1. Determine the projection of three spheres of different
radii, resting on the ground in mutual contact.
2. Determine the projections of the curve of intersection
of a cone penetrating a cylinder, the axes of the two solids
intersecting at a given angle.
3. Determine the shadow cast by the hexagonal head of
a bolt with a cylindrical shaft, the bolt standing vertically
on its screw end, from given parallel rays.
4. Determine the shadow cast by a cone standing on the
ground, the direction of the light being so arranged as to
throw part of the shadow on the vertical plane.
For shadowcasting problems, it is a good plan to place
the object in strong sunlight, so that the shadow can actually
be cast on the horizontal plane (and vertical plane, too, if
necessary), and examined. The problems then become very
300 CRAFTSMANSHIP IN MATHEMATICS
simple. Shadows cast by artificial lights are less serviceable,
since the light rays are necessarily not parallel.
As a rule there is no time for, and there is very little point
in, making projections of groups of objects, but cases of simple
interpenetrations make such good problems that one or two
are worth doing.
Speaking generally, the ground covered in orthographic
projection should be enough to enable Sixth Form boys
to solve, readily and intelligently, such stock theorems of
projection as these:
The projection on a plane of an area in another plane;
and particular cases, e.g.:
(a) Projection of an ellipse into a circle, and the ratio
of their areas.
(f$) The projective correspondence between the per
pendicular diameters of a circle and conjugate diameters in
an ellipse.
(y) Extension of the properties of polars from the circle
to the ellipse.
CHAPTER XXIII
Radial Projection
First Notions
For Sixth Form boys learning mathematics seriously, a
knowledge of radial projection is at least as important as a
knowledge of orthographic projection. Here is the sub
stance of a lesson for beginners: it is the sort of lesson one
might expect to hear an intelligent art teacher give.
Stand about 18" or 24" from a window, keeping your
head perfectly steady, and, with a piece of wet chalk, trace
accurately on the glass an outline of a distant building.
RADIAL PROJECTION 301
When you have finished, it is easy to imagine straight threads
passing from all the principal points in the building, through
the corresponding points in your sketch on the glass, to
your eye. Every line in the sketch exactly covers the corre
sponding line in the building. The drawing is another kind
of projection. But the projectors are no longer perpen
diculars; they all radiate from your eye, and they all pass
through the vertical plane on which you have made the
sketch, to the building. The vertical plane on which you
have made the picture is called the picture plane. This
kind of projection is called radial projection or perspective
projection. Perspective drawings are the kind of drawings
made by artists. Pictures are painted in accordance with
the rules of perspective. The camera also follows these
rules. Pictures and photographs represent things as they
appear to the eye.
Here is the perspective projection of a hut:
In the hut itself, the three vertical lines, AB, CD, EF, are
all equal. In the drawing, CD, the one nearest the observer,
is the longest, and those farther away are shorter. So with
the verticals in the doors and in the windows. All parallels
which recede from the observer seem to approach each other,
and at last to meet at a point on a line level with the eye.
Equal parts of a horizontal in the object are unequal in the
drawing (compare the horizontal windowbars in the two
windows). The farther away a thing is taken, the shorter it
becomes in the drawing. If you have made an accurate
chalkdrawing on the window, you can teach yourself a good
deal about perspective.
302
CRAFTSMANSHIP IN MATHEMATICS
If, however, your chalkdrawing is not satisfactory, do
this instead. Take a rectangular wooden frame of some sort
(an old pictureframe will do), say about 15" X 10". Drive
in tacks twothirds of their length, at equal distances apart,
say 1", all round the edge. Stretch cotton across the frame
and round the tacks in such a way as to divide up the frame
into squares. Now divide up a piece of drawing paper into
the same number of squares. Place the frame in a vertical
position between your eye and a suitable object or view
that may be sketched. If you sketch a house (a very suitable
object) get back far enough to see the whole house easily
within the frame. Now observe what part of the object
appears within a particular square of the frame, and sketch
that part in the corresponding square on your paper. And
so on. With care you may make a fairly accurate drawing,
and can then learn a good deal about perspective, more
particularly about converging lines and diminishing lengths.
You may also learn much from a large photograph of a build
ing, especially if you can compare the photograph with the
building itself.
Here is a perspective sketch of three bricks in a row. It
is as they would appear in a photograph. The middle brick
Fig. 138
is in the middle of the picture, and the photographer points
his camera towards it. If he were photographing, say, brick
C alone, he would turn his camera round and point towards
RADIAL PROJECTION
that brick. His picture would
then be like D. Neither A nor C
is the correct drawing of a brick
unless the brick is to the left or
right of a group of things, as in
fig. 137.
303
The Picture Plane.
Models
Use of
The ordinary perspective text
book prepared for Art teachers is,
generally, just a book of rules,
rules with only the faintest tinge
of mathematics in them. I have
known boys make faultless and
most elaborate perspective draw
ings of groups of objects in dif
ferent positions, and yet they have
had the most hazy ideas of the
inner nature of the rules they have
been applying. And yet, at bottom,
the whole thing is a study, and a
simple study, too, of similar tri
angles.
This figure shows the Picture
plane 12 ft. from the observer,
with his eye 12 ft. away and 5 ft.
above the ground at S. P. (his
Station Point). The Pictureplane
meets the ground in the ground
line. The point on the Picture
plane immediately opposite the
eye is the Centre of Vision. The
horizontal through the Centre of
Vision is called the Horizontal
Line. Radial projectors run from
304
CRAFTSMANSHIP IN MATHEMATICS
the eye to each corner of a block fixed behind the Picture
plane and cut the Pictureplane in points which, when
RADIAL PROJECTION 305
joined up, give on the Pictureplane a perspective picture of
the block.
Fig. 140 shows the sort of perspective drawing that appears
in the textbooks. The pupil must see the relation between
figs. 139 and 140. In fig. 139 the line from the eye to the
C. of V. is represented at right angles to the P.P. That line
must be supposed to be hinged at C. of V. and to turn
on the hinge through 90 until it comes into the same plane
as the P.P., as in fig. 140, which represents a drawing in one
plane, the plane of the paper. It is imperative that the boys
see fig. 139 as a model. Only then will they be able to under
stand fig. 140 completely. Then the points of distance,
vanishing points, and measuring points are all a matter of
very easy geometry.
In practice, it is an advantage to substitute for the glass
P.P. a sheet of perforated zinc, or a square of stretched
black filet net of T V' mesh, and to run threads (fastened
with drawing pins to the corner of the rectangular block or
other object being sketched) through the appropriate holes
in the zinc or net to the ring representing the eye, where
they may be secured. A drawing may then be represented
in threads of another colour, run from hole to hole in the
zinc or net, instead of in chalk as when glass is used.
Main Principles
The main principles of perspective, mathematically
considered, are all reducible to a small handful of three
dimensional problems. One will suffice to illustrate the
degree of difficulty.
Given any point on the groundplane ', to determine its
position on the pictureplane.
Since solids are determined by planes, planes by lines,
and lines by points, it will suffice to determine the position
in the pictureplane of just one point on the groundplane.
This really solves the general problem, inasmuch as any
other point may be similarly determined.
(E291) 31
306 CRAFTSMANSHIP IN MATHEMATICS
Let M be the point on the groundplane. Drop a per
pendicular MN on the pictureplane, and another EC from
the eye E to the C. of V. on the P.P. Since EC is parallel
to MN, both EC and MN, and also CN and ME, are in the
shaded oblique plane. NC is the complete projection of
the line NM extended to an unlimited length behind the
Fig. 141
P.P.; and the point R, where ME intersects CN, is the pro
jection of M. Hence R is the point to be determined.
Now in the oblique plane we have the two similar triangles
NR MN
RCE, RMN, Hence = tEl 9 i.e. CN is divided at R
. , . RC CE
in the ratio:
distance of point M from P.P.
distance of observer from P.P.
Thus CN being drawn,
R can be determined at
once.
Suppose M is 3'
behind the P.P. and the
observer is 10' in front.
It is required to divide
CN in the ratio 3 : 10.
Through C and N draw any pair of parallels. Measure off
NX equal to 3 units and CY equal to 10 units. We have
N
Fig. 142
RADIAL PROJECTION 307
. ., . , NR NX 3 . .
two similar triangles. Hence ~ = = , i.e. the
position of R in CN is determined.
Hence if the space between the two parallels represents
the P.P., if C is the C. of V., and if CN is the complete projec
tion of a line perpendicular to the pictureplane and meeting
it in N, the projection of any point in this perpendicular line
may be found by the above method.
In practical perspective we use as a pair of parallels the
horizontal eyeline and the ground line. This is a mere
matter of convenience; any other pair of parallels drawn on
the P.P. would do equally well.
Measuring points, vanishing points, and the rest are all
determined by the consideration of virtually the same prin
ciple. In fact the complete art of perspective projection
lies in that principle. With the model, the whole thing be
comes simplicity itself. The perforated zinc (or net) P.P.
with strings passing through to the eye, and the projection
of the figure threaded in with threads of a different colour,
make the main principles so clear that there is little need
for formal demonstration. The similar triangles then in
situ tell the whole story.
Sixth Form Work
When these main principles underlying the practice of
perspective have been mastered, the subject should be
followed up in the Sixth by a few of the stiffer propositions
associated with the general theory of perspective, treated
formally and deductively, more especially those concerned
with triangles in perspective, so far as these are necessary
for the understanding of the chief properties of the hexastigm
in a circle; at least Pascal's theorem should be known, though
as a mere fact in practical geometry this theorem should be
known lower down the school; its later theoretical considera
tion is always a delight to the k^en mathematical boy.
But to attack such theorems of perspective before some
308 CRAFTSMANSHIP IN MATHEMATICS
understanding of the practice of perspective has been acquired
is to attack theorems that are lifeless.
CHAPTER XXIV
More Advanced Geometry
A Possible Outline Course
What is sometimes called " Modern " Geometry or
" Pure " Geometry usually occupies a subordinate position
in Sixth Form work. This is to be regretted.
It may be readily admitted that analysis is a powerful
instrument of research, and doubtless for this reason alone
mathematicians have given it a very important place in
recent years. Accordingly, Sixth Form boys tend to devote
much time to preparation for the work of that kind which is
demanded of them at the University. But it cannot be denied
that an intimate acquaintance with geometry is only to be
obtained by means of " pure " geometrical reasoning. In
the classroom no branch of mathematics is so productive
of sound reasoning as is pure geometry. The ordinary
geometrical theorem admits of a simple, rigorous, and com
pletely satisfactory proof, a proof that is convincing and not
open to question. An elementary knowledge of the properties
of lines and circles, of inversion, of conic sections, treated
geometrically, of reciprocation, and of harmonic section,
ought to be expected from all Second Year Sixth Form boys.
Many boys now leave school without any conception of some
of the remarkable properties of the triangle and circle; and
this ought not to be.
There is much to be said for beginning with rectilineal
figures, including a fairly complete study of the tetragram
and tetrastigm, the more elementary properties of the polygram
MODERN ADVANCED GEOMETRY 309
and polystigm, and then for proceeding with harmonic section.
The remaining topics follow simply. We outline for teaching
purposes one or two of the different subjects.
The Polygram and Polystigm
These may be regarded either as systems of lines inter
secting in points, or as systems of points connected by straight
lines. The simplest figure is that determined by 3 lines or
3 points. If we have any 3 lines which are not concurrent,
or if we have any 3 points which are not collinear and which
may therefore be connected by 3 straight lines, we have
two systems which are virtually the same, and we may give
the name triangle to either.
But with more than 3 lines or points, the resulting figures
though closely related are not identical.
Rectilineal figures regarded as systems of lines are called
polygrams; as systems of points, polystigms.
A tetragram in its most general form is a complete recti
lineal figure of four lines >
no 3 of which are con
current, and no 2 parallel.
Each line is therefore in
tersected by the other 3.
If the lines be named a,
b t c, d, their points of in
tersection may be named
by combining the 2 letters
which denote the inter
secting lines. Since there ' Fig. i 43
are 3 points of intersec
tion in each of the 4 lines, we seem to have 12 points of
intersection in all, but these are reduced to 6, since each
is named twice. The 6 points of intersection are called
vertices.
A tetrastigm in its most general form consists of four
primary points, no 3 of which are collinear and which do
310 CRAFTSMANSHIP IN MATHEMATICS
not fall in pairs in parallel lines. If the points be named
A, B, C, D, their connectors may be named in the usual way,
AB, BC, &c. Since there are 3 connecting lines terminating
in each of 4 primary points, we seem
to have 12 connecting lines in all,
but these reduce to 6, since each is
named twice. The 6 lines are called
connectors.
From suitable figures, the number
of vertices and connectors in the
pentagram and pentastigm is seen to
Fig. i 44 be 10, and in the hexagram and hexa
stigm, 15.
We infer that in a polygram of n lines, and in a polystigm
of n points, the number of vertices and connectors are, re
spectively, ~F^; for the tetragram and tetrastigm give us
^~ > , the pentagram and pentastigm '^p, and the hexagram
and hexastigm *~.
In a tetragram a diagonal may be drawn from each of
the vertices to another vertex; the 6 diagonals reduce to 3.
Tetragram, with its 3 diagonals Tetrastigm, with its (4 primary and)
Fig. 145 3 secondary points
In a tetrastigm y each of the 6 connectors can intersect
another connector at a point other than the 4 primary points;
the 6 reduce to 3. These 3 new points are called the secondary
points of the tetrastigm.
From suitable figures the number of diagonals and
MODERN ADVANCED GEOMETRY 311
secondary points in the pentagram and pentastigm is seen
to be 15, and in the hexagram and hexastigm, 45.
Since
4x3x2x1, 5x4x3x2 AK 6x5x4x3
& = 15 = 45 =
8 8 8
we infer that in a polygram of n lines, and in a polystigm
of n points, the number of diagonals and of secondary points
respectively is *("i)(* 2) <"*>.
The pupil should check for the pentagram and pentastigm.
The figures for the hexagram and hexastigm are complicated
and their analysis is hardly worth while. The pupil should
note that a polygram and polystigm of the same order are
reciprocal figures; they give us an excellent example of the
principle of duality.
Derived Polygons
The pupil may be encouraged to establish, from an ex
amination of a few particular cases, the principle that the
number of derived polygons from a polygram or polystigm
is fe^L'.
For instance, the number of derived tetragons from a
tetragram or tetrastigm is 3x ^ x 1 = 3; of derived pentagons
from a pentagram or pentastigm is 4x3 * 2x 1 12; of hexa
gons, 60; and so on.
Here is a tetragram and its 3 derived tetragons:
at>
A tetragram consists of 4 lines with 6 consequent vertices,
and 3 vertices lie on each of the 4 lines. But in a tetragon
312 CRAFTSMANSHIP IN MATHEMATICS
there are only 2 vertices in a line, viz. those at the extremities
of the line; there are thus 4 vertices in all. Hence, for a
derived tetragon, we have to select 4 vertices out of the 6,
in such a way that 2, and 2 only, may lie on and determine
the extremities of each of the 4 lines. Such a selection is
known as a complete set of vertices for a derived tetragon.
Note that, whatever vertex is chosen as a startingpoint, that
vertex must be the point where the figure is completed.
Here is a tetrastigm and its 3 derived tetragons.
Fig. 147
Analogous reasoning applies. We have to select 4 connectors
out of 6, in such a way that 2, and only 2, may terminate in
each of the 4 vertices. The selection is known as a complete
set of connectors for a derived tetragon.
The boys may be given the task of drawing the 12
pentagons from a pentagram and the 12 from the pentastigm.
But they must set to work systematically or there will be
confusion. Consider the pentastigm with its 5 primary
points, A, B, C, D, E. Select AB as the initial connector.
Associated with it as a second connector we may have BC, BD,
or BE; we then have the first two connectors formed in 3
different ways, viz. AB, BC; AB, BD; AB, BE. The first
two connectors being fixed, the remaining 3 can be selected
in 2 different ways, and thus we have 6 different pentagons
formed with AB as a first connector. Now do exactly the
same thing with the other 3 connectors terminating in A.
And so on.
Do not forget that the polystigm is the key to many of
the mediaeval treeplanting problems. Given n trees, what
is the greatest number of straight rows in which it is possible
MODERN ADVANCED GEOMETRY
313
to plant them, each row to consist of m trees? For instance,
given 16 trees, plant them in 15 rows
of 4.
Construct a regular pentagram with
such of its diagonals as are necessary to
form an inner second pentagram. The
introduction of these diagonals gives 6
new points, which, with the 10 vertices
of the pentagram, make 16 points.
The general problem has never
been completely solved.
Harmonic Division *
The Pythagoras relation, golden section, and harmonic
division, are the 3 keys of pure geometry, yet harmonic division
frequently receives but very scant attention. The principle
itself once fully grasped, the actual proofs of theorems in
volving it are generally of the simplest.
The approach to the subject and its problems may be
effectively made in this way:
(1) Divide a line internally and externally in the same ratio,
Fig. 149
say 5:2. Note that the correct reading of the ratios is from
the extremities of the line to the point of division; thus for
In speaking of crossratios, avoid the term " anharmonic ", since it implies
" not harmonic ", whereas a crossratio may be harmonic, for it may be the cross
ratio of an harmonic range.
314 CRAFTSMANSHIP IN MATHEMATICS
internal division we have ~ ; for external division . Also
BPj BP 2
note the sign as well as the magnitude; e.g. for internal
division, AP l and BP X are measured in opposite directions
and the ratio is therefore negative; for external division
AP 2 and BP 2 are measured in the same direction, and the
ratio is therefore positive. Give important examples of this,
for instance the theorems of Ceva and Menelaus.
(2) Algebraic Harmonic Progression. Definition: If a,
, , . TT , a b a 1.12
0, and c are in H.P., then = ; or,  +  ; or
b c c a c b
h
a f c'
(3) Compare the geometry and the algebra. A line AQ
is said to be " harmonically divided " at P and B when, if
AQ = a, AB = b 9 AP = c, a, b, and c are in H.P.
c . a  b a /t , c . . , . BQ AQ
Since =  (by definition), /._=:_,
b C C JjJr Ar
or AQ X BP = AP X BQ; (Cf. fig. 150.)
i.e. product of whole line and middle segment equals product
of external segments. Hence if AB is divided harmonically
A I 1 ! 19
^ .jaJ *,
j^_ b 1 ^j
Kc H
Fig. 150
at P and Q, PQ is divided harmonically at A and B. In other
words, AB is divided internally and externally at P and Q
in the same ratio; and PQ is divided internally and externally
at B and A in the same ratio.
(4) Harmonic Ranges. If a line AB is divided harmonically
at P and Q, the range of points {AB, PQ} is called a harmonic
" range ". The pair of points A and B are said to be con
jugate to each other; so with the points P and Q. We may
MODERN ADVANCED GEOMETRY 315
conveniently name a harmonic range thus (AB, PQ}, the
comma being inserted to distinguish the pairs of conjugate
points.
(5) Harmonic Pencils. Define " ray " and " pencil ".
Every section of a harmonic pencil is a harmonic range, e.g.
(AB, PQ), (A'B', P'Q'). A pencil O.APBQ is harmonic if
C> Q
Fig. 151
a transversal MN parallel to one ray OQ is bisected by the
conjugate OP.
A range may be read {AB, PQ} or (APBQ), and a pencil
may be read O(AB, PQ) or O.APBQ. Adopt one plan and
adhere to it, or the boys may be confused. It is a good plan
to use coloured chalks for every harmonically divided line
on the blackboard, and always of the same colour. Harmonic
division is so useful that its immediate recognition is desirable.
The Complete Quadrilateral
The commonest theorems involving harmonic section
concern (1) the complete quadrilateral, (2) pole and polar.
Fig. 152 (i) shows a tetragram with its 3 diagonals (2 pro
duced to meet) which are indicated by heavy lines. Each of
the 3 diagonals is harmonically divided by the other two. Fig. (ii)
shows a tetrastigm with its 6 connectors also indicated by
heavy lines, and with lines (faint) joining the secondary
points in pairs. Each of the 6 connectors is harmonically divided
by (1) the secondary point through which it passes, and (2)
the line joining the other two secondary points. If we superpose
one figure on the other, we get a remarkable series of harmonic
316 CRAFTSMANSHIP IN MATHEMATICS
pencils, of mutual harmonic intersections, and of collinearities
Actually, of course, they can be picked out in fig. 152 (ii).
(0
Fig. 152
If the complete quadrilateral is approached in this way,
the boy's interest and curiosity is aroused. He is greatly
surprised to discover that a simple thing like a quadrilateral
has so many remarkable properties.
But the important thing is for the boy to realize that this
general quadrilateral generalizes the theorems of all particular
quadrilaterals. For instance, suppose that the line EF in the
last figure is removed to an " infinite " distance from C, the
4 points C, A, D, B, become the vertices of a parallelogram;
and since R is the harmonic conjugate of the point in which
CD intersects EF with respect to the points C and D, R
becomes the middle point of CD. Thus the theorem of
the complete quadrilateral is a generalization of the theorem
that the diagonals of a parallelogram bisect each other.
Impress on the boys the importance of the use of a general
figure in their geometrical work, and the fact that from its
properties the special properties of a particular figure may
often be inferred.
Pole and Polar
The dark lines in the figures represent harmonically
divided lines: (i), (ii), every chord which passes through
the pole P is cut harmonically by the polar; (iii), if the polar
of PX passes through P 2 , the polar of P 2 passes through P x .
In all such cases, let the pole and the corresponding point
MODERN ADVANCED GEOMETRY
317
on the polar be consistently marked, say, by P's. Also let the
circumference be cut in Q's. A consistent system of naming
all harmonically divided lines is a great advantage.
(0
For a good general problem on harmonic section, see
Scientific Method, pp. 3879.
Concurrency and Collinearity
Many theorems involving these principles form excellent
practical problems for careful work in junior classes. En
courage young boys to bisect the angles of a triangle, to bisect
the sides, to draw perpendiculars from the midpoints, &c.,
and to make discoveries for themselves. Let them thus obtain
the facts. A little later, the simpler theorems and their proofs
may be given; e.g. concurrent lines through the vertices
of a triangle, the medians, the perpendiculars to the opposite
sides, two exterior bisectors and the internal bisector of the
three angles. A little later still: given the Menelaus relation,
prove that the points are collinear; given the Ceva relation,
prove that the points are concurrent. Pascal's and Brianchon's
theorems will, of course, always be included. Another good
type of theorem is this: four points on a circle and the tangents
at those points form, respectively, two quadrilaterals whose
internal diagonals are concurrent and form a harmonic pencil,
and whose external diagonal points are collinear and form
a harmonic range. The principles of concurrency and col*
linearity are so important that they cannot be too strongly
318 . CRAFTSMANSHIP IN MATHEMATICS
emphasized, but with most boys facility comes only after
much practice with varied types of problems.
Pascal's theorem suggests the study of the hexastigm, of
which that theorem is the simplest property. The theorem is
usually quoted, " The opposite sides of any hexagon in
scribed in a circle intersect in 3 collinear points ", but a
more precise statement is, " The 3 pairs of opposite connectors
of a hexastigm inscribed in a circle intersect in 3 collinear
points ". Fully expressed, this comes to, " The 15 connectors
of a hexastigm inscribed in a circle intersect in 45 points
which lie 3 by 3 on 60 lines ". I have seen one passable
figure prepared by a boy; he was looked upon as the fool of
his Form, though he was extraordinarily successful in the
use of ruler and compasses. Elaborate drawing of this kind
is largely a waste of time, and, after all, the hexastigm still
remains to be investigated fully.
The Further Study of the Triangle and Circle
There are numerous theorems on the triangle, many of
them simple, many useful, many beautiful. For instance,
those concerning triangles in perspective, pole and polar with
respect to a triangle, symmedian points of a triangle, Brocard
points of a triangle.
So with the circle: the ninepoint circle, escribed circles,
the cosine circle, the Lemoine circle, the Brocard circle.
It is important to leave on the boy's mind a vivid im
pression of the remarkable properties that even now are
frequently being discovered concerning the triangle and
circle. Boys who, when they leave school, know no more
pure geometry than that contained within the limits of
School Certificate requirements are certainly not likely to
devote leisure moments to a systematic playing about with
circles and triangles, in the hope of hitting on some new and
perhaps remarkable property yet undiscovered.
MODERN ADVANCED GEOMETRY 319
Conic Sections
The pure geometry and the algebraic geometry of the
cone should be studied side by side. If either has to be
sacrificed, let it be the latter. Algebraic manipulation is all
very well, but the cone is a thing which occupies space, and
when its spatial relations are reduced to symbols, these
symbols may assume, in the pupils' minds, an importance
which is not justified, and the geometry proper may be
overshadowed.
CHAPTER XXV
Geometrical Riders and their Analysis
In those schools where riders are, as a rule, solved readily,
schools where boys take a real delight in attacking new ones,
the secret of success seems to be that right from the first
every new theorem and every new problem has been presented,
not as a thing to be straightaway learnt, but as a thing to
be investigated and its secret discovered. The boys do not
learn a new theorem or problem until they have been taught
how to analyse it, and to discover how it hangs on to what
has gone before.
General instructions should include advice as to the
necessity of drawing a general figure, of drawing that figure
accurately, and of setting out definitely what is " given "
and what is to be proved. We append a few instances of
problems and theorems actually solved in the classroom,
with a brief summary of the sort of arguments used.
1. O is the midpoint of a straight line PQ, and X is a point
such that XP = XQ. Prove that the Z.XOP is a right angle.
320 CRAFTSMANSHIP IN MATHEMATICS
We argue in this way:
(1) What facts are given?
OP = OQ,
XP = XQ.
(2) What have I to prove?
That ZXOP is a right angle.
(3) Since I have to prove that Z.XOP is a rt.
join XO.
x
I must
(4) How have I been able to prove before, that an L is
a it. Z?
(i) Sometimes by finding it to be one of the two eq.
adj. Z.s making a str. L.
(ii) Sometimes by finding it to be an angle in a semi
circle.
(iii) Sometimes by finding it to be at the intersection
of the diag. of a sq. or a rhombus,
(5) The first of these looks possible here. Are the adj.
/.s at O equal? Yes, if the two As XOP and
XOQ are congruent.
(6) Are these As congruent? Yes, three sides in the one
are equal to three sides in the other, as marked.
Now I know how to write out the proof, in the ordinary
way: I therefore begin again, and make up a new figure as
I proceed.
GEOMETRICAL RIDERS 321
Proof. Join XO. In the As XOP, XOQ,
OP == OQ, (given)
XP  XQ, (given)
XO is common,
.. A XOP = A XOQ; (three sides)
i.e. the two As are equal in all respects.
.. ^XOP ZXOQ,
. . ^XOP == 90. (half the str. POQ)
(which was to be proved)
2. ABCD is a parallelogram; E is the midpoint of BC,
and AE and DC produced intersect at F. Prove that AE = EF.
Argument.
(1) What facts are given?
(i) ABCD is a /Z7m; . . its opp. sides are .
(ii) BE == EC (by constr.).
(2) What have I to prove?
That AE  EF.
(3) How have I been able to prove before, that two lines
are equal?
(i) Sometimes by finding them in two congruent As.
(ii) Sometimes by finding them in a A with two angles
equal,
(iii) Sometimes by finding them to be the opp. sides
of a H7m.
(B291) 22
322
CRAFTSMANSHIP IN MATHEMATICS
(4) Does either of these plans seem possible, to prove
AE eq. to EF?
(5) Yes, the first, for the As ABE and FCE look congruent.
(6) Are they congruent?
(7) Yes. Two Zs and a side, as marked.
Now I know how to write out the proof in the ordinary way
Proof. In the As ABE, FCE,
BE = EC,
ZAEB = ZFEC,
ZABE = ZFCE,
A ABE A FCE,
.. AEEF.
(constr.)
(vert. opp. Ls)
(alt. Ls; BC across \\s AE, DF)
(2 Z.s and a side)
(which was to be proved)
3. Draw a circle of " radius to touch the given line AE
and the given circle CDE. (The given line must not be more
than 1" from the given circle.)
(*>
Fig. 156
This is a problem, and we have to discover the method
of construction. I assume the problem done, and I make a
sketch of the required circle in position, as accurately as
possible (fig. 156, i). I examine the figure, and I observe that
the line QR to the pt. of contact R = tf and is AB; that
HQ = HP + Y and passes through the pt. of contact P.
Argument.
(1) Since the required O has to touch the line AE> its
GEOMETRICAL RIDERS 323
centre must lie somewhere on a line FG  AB, and
Y from AB (fig. 156, ii).
(2) Since the required O has to touch the O CDE, its
centre must lie somewhere on the circle LMN
having the same centre as CDE and having a radius
HQ equal to radius HP + (fig. 156, ii).
(3) Since the centre of the required O lies both on the
line FG and on the circle LMN, it must be at a
point of intersection of FG and LMN.
Now I know how to construct the circle.
Construction.
(1) Draw a line FG  AB, " away from it.
(2) From centre H, with radius equal to HP + i"> draw
O LMN.
(3) From one of the pts. of intersection of this line and
circle, say Q, as centre, draw a circle RSP of " radius. This
is the required O.
Proof.
(1) The circle RSP has a radius of ". (constr.)
(2) The circle touches AB (in R), for any circle of \*
radius having its centre on the line FG must touch AB, a
from the centre Q passing through the pt. of contact.
(constr.)
(3) The circle touches the given circle CDE (in P), for
any circle of J" radius having its centre on the circle LMN
must touch CDE, the line joining the centres passing through
the point of contact, (constr.)
(4) Therefore the circle is constructed in accordance
with the given conditions, (which was to be done).
4. From the right angle of a rightangled triangle, one
straight line is drawn to bisect the hypotenuse, and a second
is drawn perpendicular to it. Prove that they contain an angle
324 CRAFTSMANSHIP IN MATHEMATICS
equal to the difference between the two acute angles of the
triangle.
A
BED c
Fig. 157
Given.
(1) a rt. L at A.
(2) rt. Zs at E.
(3) DB = DC.
(4) DA  DB DC. (Since a O will go round the
A ABC on BC as diameter.)
Further, an examination of the figure shows that in the
2 rt. Zd As BAC, BEA, ZABE is common; .'. the two As
are equiangular. This fact may prove useful.
Required to prove: ZEAD ( ZABC  ZACB).
Argument. This is a type of problem in which we may
first usefully test a particular case by assigning to some
angle a number of degrees, and then calculating the
number in some or all of the other angles. For instance,
let ZABC 65 (not 30, or any other factor of 360, lest
a fallacy creep into our argument).
If ZABC = 65, ZACB = 25 (the complement).
If ZACD  25, ZCAD  25 (for AD  DC).
Also ZBAE = 25 (equiangular As, as above).
Again, if ZABC = 65, ZBAD = 65 (for DB = DA),
and ZEAD == ZBAD  ZBAE  65  25 = 40.
But ZABC  ZACB  65  25  40,
/. ZABC  ZACB = ZEAD.
GEOMETRICAL RIDERS 325
Thus the theorem is true in this particular case. We are
therefore now in a position to generalize the result and to
set out the proof in the ordinary way.
Proof.
(1) DB = DA, (given)
.. ZDBA ZDAB.
(2) BAG and BEA are equiangular rt. Zd As. (given)
.. ZACB ZEAB.
(3) .. ZDBA  L ACB = ZDAB  ZEAB (from 1 and 2),
 ZDAE.
(which was to be proved)
5. The figure shows an equilateral triangle ABC within
a rhombus ADEF, a side of the former being equal to a side
of the latter. Determine the magnitude of the angles of the
rhombus.
Argument.
From an examination of the figure we know the following
facts:
(1) Rhombus: 4 equal sides; opp. sides ; opp. 2Ls equal.
(2) Equil. A : sides equal; Z.s equal.
(3) As ADB, AFC isosceles.
(4) A EEC isosceles (by symmetry).
326 CRAFTSMANSHIP IN MATHEMATICS
All that we know about L magnitudes from the figure are:
(1) Anglesum of any A = 180.
(2) Each L of A ABC = 60.
We have therefore to try to express the Zs of the rhombus in
terms of these values.
From the rhombus, ZFAD + ZADB = 2 rt. Zs.
Also, ZABD + ZABE = 2 rt. Zs.
But ZADB = ZABD, (isos. A)
.. ZFAD ZABE.
Obviously, therefore, ZFAD ZFED ZABE /ACE
Now the sum of the last 3 of these Zs
= (sum of Zs of A EEC) + ZABC + ZACB
= 180 + 120
 300.
= 100;
.. ZADE = (180  100) = 80.
The estimate may be set out formally in almost the same
sequence.
6. Show that the 4= straight lines bisecting the angles of
any quadrilateral form a cyclic quadrilateral.
Fig. 159
GEOMETRICAL RIDERS
327
Let ABCD be the given quadrilateral, and let the bisectors
of the Zs form the quadrilateral EFGH.
Let the Zs be marked as shown.
Given: a = a'; j8 = j8'; y = y'; 8 == 8'.
Argument,
If a circle will go round EFGH, the sum of any 2 opp.
Zs of EFGH 2 rt. Zs; thus a + T = 2 rt. Zs.
// a + r = 2 rt. Zs, a + j8' + y + 8' = 2 rt. Zs since
the sum of all the As of the 2 As ABE and CDG
= 4 rt. Zs.
But a + /J' + y + 8' we &wo; are equal to 2 rt. Zs, for
2a + 2)3' + 2y + 28' = 4 rt. Zs (the 4 Zs of the
quadl.)
Thus we have found the key.
Proof.
The sum of the 4 Zs of the quadl. ABCD = 4 rt. Zs.
. . the sum of the halves, a + ft' + y + 8' = 2 rt. Zs.
But the sum of all the Zs of As ABE and CDE = 4 rt. Zs.
. . a' + T' = 2 rt. Zs,
..a + T = 2 rt. Zs,
.. the points E, F, G, H are concyclic.
(which was to be proved)
7. Three points D, E,
and F in the sides of a tri
angle ABC are joined to
form a second triangle, so
that any two sides of the
latter make equal angles with
that side of the former at
which they meet. Show that
AD, BE, and CF are at
right angles to BC, CA, Fig . I6o
and AB, respectively. (You
may not assume properties of the pedal triangle.)
328 CRAFTSMANSHIP IN MATHEMATICS
Given: a = a'; = '; y = y'.
Required to prove: AD is L to BC, &c.
Argument. Assume that AD is L to BC.
Then .* a = a', S = S'.
DF FG
=
FG GE'
DA
But this is known, since each ratio =
AG'
LThis is easily shown: Produce DE to H; EA is the
DF DA
bisector of the external ZGEH of the A GDE; ..== = .
EG AG
Similarly, by producing DF to K, it is seen that  . = _ .
AG J
Thus we can make these known ratios our startingpoints,
and set out the proof in the usual way.
Proof.
Produce DE to H and DF to K. Then EA and FA are
the bisectors, respectively, of ext. ZGEH of A GDE, and
of ext. dGFK of AGDF.
DE DF , , DA
" EG = FG' f r CaCh = AG'
DE = EG
" DF FG'
/. DG bisects ZEDF,
.'. 8 = 8',
/. 8 + a = 8' + a',
.. AD is to BC.
GEOMETRICAL RIDERS
329
Similarly, we may show that BE L AC, and CF JL AB.
(which was to be proved)
8. Show that the perpendicular drawn from the vertex
of a regular tetrahedron to the opposite face is 3 times that
drawn from its own foot to any of the other faces.
Let ABCD be the tetrahedron, and let AE be the L from
the vertex A to the opp. face BCD. Then E is the centroid
of the ABCD.
Let a L EF be drawn to the face ACD; F will meet the
median AG.
SECTION THROUGH TETRAHEDRON
Fig. 162
To prove: AE = 3EF.
Argument: Consider the vertical section through ABG.
We know that, since E is the centroid of BCD, EG = BG,
Is there an analogous relation between EF and AE?
If we draw BK 1 AG in the face ACD, BK must be
equal to AE.
E F _ EG = i
BK " BG 3 '
/. EF = JBK = AE
For the other faces similar results follow from symmetry.
330 CRAFTSMANSHIP IN MATHEMATICS
Proof.
Let a 1 BK from B meet the median AG in K; BK = AE.
EG = BG. (E is the centroid of BCD)
EF _EG ,
BK BG *'
/. EF  PK
= JAE. (which was to be proved)
If a rider is in any way of an unusual character, pupils
sometimes have difficulty in writing out a proof concisely.
We give an example of an acceptable proof for such a rider.
Fig. 163
In a given triangle ABC, BD is taken equal to onefourth
of BC, and CE equal to onefourth of CA. Show that the
straight line drawn from C through the intersection F of BE
and AD will divide the base into two parts at G which are
in the ratio 9 to 1.
ABEA3 A EEC,
AFEA = 3 AFEC,
.'. ABFA=3 ABFC,
= 12 ABFD,
.'. AF = 12 FD,
/. AAFC= 12 ADFC
= 36 ABFD
= 9 ABFC.
GEOMETRICAL RIDERS 331
Now the As AFC, BFC are on the same base FC. Hence the
vertical height of A AFC above this base = 9 times the
vertical height of A BFC above this base.
.'. A AGC 9 ABGC, (on the extended base, GC)
AG = 9 GB.
Any reasonable examiner would accept a proof given in
this form and would be glad to be saved from the trouble of
reading defensive explanatory matter.
Books on geometry to consult:
1. Plane Geometry, 2 vols., Carson and Smith.
2. Geometry, Godfrey and Siddons.
3. Geometry, Barnard and Child.
4. Elementary Concepts of Algebra and Geometry, Young.
5. Elementary Geometry, Fletcher.
6. Cours de Ge'ome'trie, d'Ocagne (Gauthier Villars).
7. A Course of Pure Geometry, Askwith.
8. Modern Pure Geometry, Lachlan.
9. Sequel to Elementary Geometry, Russell.
10. Geometry of Projection, Harrison and Baxandall.
11. Protective Geometry, Matthews.
12. An Elementary Treatise on CrossRatio Geometry, Milne.
13. Foundations of Geometry, Hilbert.
14. The Elements of NonEuclidean Geometry, Sommerville.
15. Space and Geometry, Mach.
16. Analytical Conies, Sommerville.
17. Curve Tracing, Frost (new edition). An old and faithful friend.
18. Euclid, 3 vols., Heath. The work on the subject.
332 CRAFTSMANSHIP IN MATHEMATICS
CHAPTER XXVI
Plane Trigonometry
Preliminary Work
The pupils' first /acquaintance with the .tangent, sine,
and cosine should be made during their elementary lessons
in geometry. Boys soon (learn that the symbols for the
trigonometrical ratios^ may enter into formulae which can
be manipulated algebraically; and since, in the algebra
course, the study of x n and a x is included, it is difficult to
exclude from it the study of sin x and tan x. Each represents
a typical kind of function. To each corresponds a specific
form of curve its own particular picture, the graphic picture
of the function. Algebra and trigonometry should be much
more closely linked together, and much of the purely formal
side of trigonometry might with advantage be sacrificed,
and greater stress be laid on the practical and functional
aspects of the subject. The needs of coordinate geometry
and the calculus, of mechanics and physics, should always
be borne in mind; in fact, much of the work done in trigo
nometry might be directed towards these subjects.
The notion of an angle as a rotating line should be given
at the very outset of geometry, so that, when in trigonometry
angles greater than 180 are discovered, the notion will
already be familiar. The angle of " one complete rotation ",
and its subdivisions, straight angle, right angle, and degree,
will, of course, be known, and pupils should be able to draw
freehand, at once, to a fairly close approximation, an angle of
any given size, the 30, 45, and 60 angles being quite
familiar from the half equilateral triangle and the half square.
Coordinate axes and the four quadrants will also be
familiar from previous work on graphs; so will directed
algebraic numbers. Angles of elevation and depression will
already have been measured in connexion with practical
PLANE TRIGONOMETRY
333
problems in geometry and mensuration. Pythagoras should
be at the pupils' fingerends; so should the fundamental
idea of projection} Similar triangles should also be known,
and ratios of pairs of sides should be equated with readiness.
Unless all these things are known, really known, the earlier
work in trigonometry is much hampered by timeconsuming
preliminaries.
Do not scare the class in the first lesson by hurling at
their heads all six trigonometrical ratios. Only the tangent,
sine, and cosine need be studied at first, and these one at a
time, each as a natural derivative of practical problems of
some kind.
The Tangent
The tangent should come first. Revise a few simple
geometry problems in heights and dis
tances, and let the new trigonometrical
term gradually replace the geometrical ratio
which the boys already know.
We might begin in this way.
Measure the height of the school flagstaff
AB.
Set up the 4' high theodolite at D, at
a distance of, say, 25' from B, and measure
the angle AEC (== 58). Make a scale
drawing. By scale, AC = 40'. Hence AB
= AC + CB = 40' + 4' = 44'.
Thus the ratio  ~ = 16.
In other words, when the angle E is 58, AC = 16 EC.
Now look at a series of rightangled triangles with the base
angle 58. In every case the ratio AC/CE is the same, since
the triangles are similar. Thus in each case AC = 16 EC.
Hence, whatever the length of EC, we can find the length
of AC by multiplying EC by 16. (Fig. 165.)
334
CRAFTSMANSHIP IN MATHEMATICS
Thus the number 16 is evidently associated with the
particular angle 58. How? It measures the ratio AC/CE,
i.e. the P^Pendicular of the rightangled triangle AEC. If,
base
then, we make a note of this value 16, as belonging to the
particular angle 58, we are likely to find it very valuable
when dealing with rightangled triangles having an angle
E c E c E c
Fig. 165
of 58; if we know the base we have merely to multiply it
by 16 to obtain the perpendicular.*
Obviously every angle, not merely 58, must have a special
value of this kind. We may take a series of rightangled tri
angles, with different base angles, say 10, 20, 30, 40, 50,
60, 70, 80, measure their perpendiculars and bases to scale,
calculate their ratios, and make up a little table for future use.
If we liked, we could draw these triangles independently,
though that would make the arithmetic rather tedious. An
easier way is to draw a base of exactly 1" in every case; then
our arithmetic is easy (fig. 166). (Any number instead of 1
would do, but that would mean a little more arithmetic.)
58;
EC 36
Do not mention the term hypotenuse at all.
dealt with.
Let that wait until the sine
PLANE TRIGONOMETRY
835
Mathematicians sometimes make the perpendicular a
tangent to the circle, fig. 167 (they always remember that
an angle is concerned with rotation): and for convenience
they call the ratio P er P^ ndlcu _ lar the tangent of the angle.
base
Thus they say, tangent 10 == 18; tangent 20 = 36; and
so on. They generally write tan for tangent.
Bi
rj
84 ;
Fig. I 66
Fig. 167
But remember that the tangent of an angle is just a number
which shows how many times the perpendicular is as
perp.
long as the base; in other words, it is the ratio
Since
perp.
^*  tan,
base
perp. = base X tan\
base
hence in the
triangle ABC, AC = BC X tan 35,
i.e. the tan of an angle is the mul
tiplier for converting the base into
the perpendicular. (Fig. 168.)
There are better ways of finding
these values than by merely drawing
to scale; in fact, values to 7 places of
decimals have been found, the work
to be done with them (by surveyors,
for instance) having often to be very accurate. Here is a little
Fig. 1 68
336
CRAFTSMANSHIP IN MATHEMATICS
table giving the values of the tangents of 10 angles, to 4 places
of decimals.
tan 10=
20 =
30 =
40 =
1763
3640
5774
8391
45 = 10000
tan50= 11918
60 = 17321
70 = 27475
80  5671
89 = 5729
373
y
/
/
c XT
y
/
"for
75
373
__/
/
/

^^*
^
/
, 
^
^ '
^ '
l(
5* 2
3
0' 4
A
0* 5
<*
7
r a
75
0* 9C
Fig. 169
There is no tangent for 90. Can you see why? Can you
see why the tan of 89 is so large? look at fig. 167. Can
you see why the tan of 89 59' 59" must be enormously
large?
You will remember how, when we had graphed a
function of #, we were able to obtain other values by
interpolation. We may do the same with the tan graph;
in fig. 169, plotted from the above table, you may see
that the tan of 75 is about 373. To get anything
like accurate values, we should have to have a very large
graph.
We give one or two easy practical exercises.
PLANE TRIGONOMETRY
A ladder leaning against a house makes an
angle of 20 with the wall. Its foot is 10' away.
How high up the house does it reach?
We have to obtain the height AC, and we
therefore require to know the tan of the angle B.
Since A  20, B = 70.
= tanB = tan 70
BC
= 27475 (see table or graph).
AC ^ BC X 274:75
 10' X 27475
= 27475'.
Fig. 170
Two boys are on opposite sides of a flagstaff 50' high.
Their angles of elevation of the top of the staff are 20 and
30, respectively. How far are they apart?
Fig. 171
Given, length of AB; Required, length of BC and BD.
Since the angles at C and D are given, we may mark in the
angles at A.
Distance of boys apart CD
= CB + BD
= AB tan 70 + AB tan 60
= 50(27475 + 17321)
 22398 (feet).
Give ample practice in easy examples of this kind until the
boys are thoroughly familiar with the fact that the tan is
just a multiplier, sometimes less than 1, sometimes greater,
(B291) 23
338
CRAFTSMANSHIP IN MATHEMATICS
for calculating the length of the base from the per
pendicular. Vary the exercises, so that the base is not
always a horizontal.
The Sine
To beginners, navigation problems for introducing the
sine seem to be a little difficult, and may best be taken a little
later. Here is a suitable first problem. A straight level road
AB, 20 miles long, makes an angle of 37 with the westeast
direction AC. How much farther north is B than A?
In the figure we have to find the length BC. It is easy to
find this length from a scale drawing: BC =12 miles, i.e.
B is 12 miles north of C.
Fig. 172
G
Fig. 173
BC
BA*
Now examine the ratio ^. As long as the angle A in
a rightangled triangle remains 37, the ratio must always
be the same, no matter what the length of the sides, e.g.
BC DE FG T  k f r ,.
p . If then we know the value of this ratio
for one triangle, we know it for all similar triangles; its
value is ^j r * 6  Thus, if AD = 14, DE = 6 of 14 = 84;
and so on.
This new ratio is Perpendicular and is called sine. It
hypotenuse
is a mere number, and represents how many times the per
pendicular is as long as the hypotenuse. We ought really
PLANE TRIGONOMETRY
339
to say, represents what fraction the perpendicular is of the
hypotenuse, since the value is always less than 1. Thus
sine 37 = *6 (we generally write sine, sin, though we pro
nounce " sin " as " sine ").
Just as with the tangents, so with the sines: we might
draw a series of rightangled triangles with base angles
successively equal to, say, 10, 20, 30, &c., and so construct
a table. When we constructed fig. 166 for the tangents, we
made a triangle with a base of 1 unit, because we wanted to
Fig. 174
make the arithmetic easy, and then the base was the de
nominator of the ratio. In the case of the sine, we will also
make the denominator of the ratio unity, i.e. we must now
make the hypotenuse unity. Here is a plan for doing this.
With O as centre, and unit radius, draw a circle. With the
protractor, mark in the angles 10, 20, 30, &c.; each radius
OB, OC, &c., is equal to unity. From the ends B, C, D, &c.,
of these radii, drop perpendiculars to the base, BG, CN, DK,
&c., and measure them. Since OA = OB = OC (&c.) = T,
the perpendiculars will be fractions of 1". Now we may
* . BG 17 . n CN
obtain the sines: smlO = ~  == = 17; sm20 = r
34
= = 34; sin 30 = '50, &c. By careful measurement, we
340
CRAFTSMANSHIP IN MATHEMATICS
may obtain sines to 2 decimal places. Here is a little table
to 4 places.
sin 10 =
20 = 
30 =
40 = 
45 =
(,
1736
3420
5000
6428
7071
si
i
n50
60 C
,70 f
, 80 C
, 90 C
D
5
)
)
) '
766
866
939
984
LOOOi
o
,
^
'"'"
9
Sr>
55
1
82
/
S
.0
/
^
f
,
/
^iKiee. ^?
/
/
/
.X
,
/
/
y
1
/
10" 20* 30* 4CT 50* 60* 70* 80* 9O*
65
ANGLES
Fig. 175
By drawing the sine graph, we may obtain the sine of any
A other angle up to 90, by inter
polation; e.g. sin 55 is about 82.
Remember that the W of an
angle is just a number. Since
perpendicular _ ^ . _
hypotenuse
dicular = hypotenuse X sine. Hence
Fig. .76 " inthetriangleABC,AC=ABsin35,
PLANE TRIGONOMETRY
341
i.e. the sine of an angle is the multiplier for converting the
hypotenuse into the perpendicular. In this case the multiplier
happens to be always a fraction.
Here are one or two easy typical problems:
A ladder 30' long stands against a vertical wall. It makes
an angle of 70 with the ground. What is the height above
the ground of the top of the ladder? (Fig. 177.)
Given, AB = 30'; ZABC = 70. Required AC.
AC
= sin 70 = 94 (from table or graph),
A. 5
/. AC = AB x 94 = 30' x 94 = 282'.
A railway slopes at an angle of 10
for a distance of 1000 yards. What is the
difference in level of its two ends? (Fig.
178.)
Fig. 178
Given, AB = 1000 yards; Z.ABC = 10. Required AC.
AC
AB
= sinlO = 1736.
/. AC = AB X 1736 = 1000 yd. X 1736 = 1736 yd.
The Cosine
Projection problems form a suitable beginning. AB
represents a sloping road 500 yd. long. A surveyor finds that
it makes an angle of 30 with the horizontal What is the
342
CRAFTSMANSHIP IN MATHEMATICS
projected length on a horizontal line, such as would be shown
on an ordnance map?
The projection of a line AB on another line MN is the
distance between two perpendiculars drawn to MN from the
M
Fig. 179
N
ends of AB. If MN passes through A, one end of the road,
only one perpendicular (BC) is necessary. The projection is
then AC.
Given, AB = 500 yd.; ZBAC = 30. To find AC.
From a scale drawing we find that AC 433 yd.
AC
Now examine the ratio As long as the angle A in a
AJJ
rightangled triangle remains 30, the ratio must always be
B
A G E C
Fig. 1 80
the same, no matter what the length of the sides, e.g. = 
A /* AJb> AD
= pgr, for the triangles ABC, ADE, AFG are all similar. If
then we know the value of this ratio for one triangle, we know
433
it for all similar triangles. Its value is or 866. This
base 50
new ratio, ^ tenuse > is called the cosine (generally written
PLANE TRIGONOMETRY
base
343
cos). It is a mere number. Since f = cos, base = hypo
hyp.
tem/5* X cos. Hence, in the triangle ABC, AC = AB cos 30,
i.e. the cosine of an angle is the multiplier for converting
the hypotenuse into the base. In this case, again, the
multiplier always happens to be a fraction.
Nf
Just as with the tangent and sine, so with the cosine:
we may draw a series of rightangled triangles with base
angles successively equal to say 10, 20, 30, &c., measure
them up, and so construct a table. And as in the case of the
sine, we will so construct our triangles that the length of the
hypotenuse is always unity.
344 CRAFTSMANSHIP IN MATHEMATICS
Here is a little table of cosines, to 4 places of decimals:
cos 10 = 9848
20 = 9397
30 = 8660
40 = 7660
45 = 7071
cos 50 = 6428
60 = 5000
n 70 = 3420
M 80 = 1736
90 =
By drawing a cosine graph from the above values, we can,
by interpolation, obtain the value of any other angle up to
90, e.g. cos 35 = 82 (approx.).
COSINES '
Q
" **^
"\
^
".A"
SPS,
_3S =
_Q2
\
O
.7
'
\
./
\
\
\
4
\
.X
\
,
2
\
\
\
l(
J Z
c5
4
G
7
kO 9(
55"
DEGREES
Fig. 182
Compare the sine and cosine graphs. Each is an exact
lookingglass reflection of the other. Now look at che two
tables of sines and cosines. Each is the other turned upside
down. Evidently there is a curious connexion between sines
and cosines.
It is easy to draw both sine and cosine curves by means
PLANE TRIGONOMETRY
345
of intersecting points made by (1) parallels from an angle
divided quadrant, and (2) perpendiculars from the corre
spondingly divided abscissa. Note how the two curves
together form a symmetrical figure, and how they cut in
one point. What do you infer about this point common
\
O 10 20 30" 40*. 50" 60* 70* 60* 90
4*
Fig. 183
to the two curves? There is evidently some angle the sine
and cosine of which have the same value. Look at the two
tables.
Easy cosine problems. (1) The legs of a pair of compasses
are 5" long. Find the distance between the points when the
legs are opened to an angle of 80.
C
Fig. 184
Given: AB = AD = 5"; /.BAD = 80. If AC is the bi
sector of /.BAD, ^BAC = 40; hence /.ABC = 50.
346 CRAFTSMANSHIP IN MATHEMATICS
Required: length of BD (= 2BC).
Dp
r = cos 50; /. BC = AB cos 50
= 5" X 64,
.'. BD = 10" x 64 == 64".
(2) C is any point in the line XY. CA and CB are drawn
on the same side of XY so that CA = 4", CB = 5", LXCA
= 40, L YCB = 60. Find the projection of ACE on XY.
^ M C N T
Fig. 185
Drop perpendiculars AM, BN, on XY. Then the projec
tion of ACB on XY is MN. Required: the length of MN.
MN = CM + CN
= AC cos 40 + BC cos 60
= (4" X 77) + (5" X 50)
= 558".
Now give the boys the same two problems again, making
them use the sine instead of the cosine. Hence give them
the first notion that the sine and cosine are so closely related
that one may sometimes be used instead of the other. Make
them remember this:
If the hypotenuse is given,
(1) use the sine to find the perpendicular;
(2) use the cosine to find the base.
PLANE TRIGONOMETRY
347
The sin, cos, and tan: Simple Interrelations
Introduce the notation a, b y and c to represent the number
of units of length in the sides opposite the angles corre
spondingly named. Also show that since Z.A + Z.B = 90
A = 90 B, and B = 90 A. Now tabulate:
 = tanB, or b = a tanB;  = tanA, or a b tanA.
a b
 = sinB, or b = c sinB;  = sinA, or a = c sinA.
c c
_ = cosB, or a = c cosB;  = cosA, or b = c cosA.
c c
Hence,
(1) Since ? = tanA, and  = tanB, /. tanA =
b a
(2) Since  = sinA = cosB,
(3) Since  = cos A = sinB,
c
w l = b'
c
(5) Similarly,
tanB
sinA = cosB.
cosA = sinB.
sinA
cos A
sinB
= tanA.
tanB*
348
CRAFTSMANSHIP IN MATHEMATICS
(6) Since B = 90 A, and sin A = cosB,
.*. sinA = cos(90 A).
(7) Since B = 90 A, and cos A = sinB,
/. cos A = sin (90 A).
All these relations must be carefully committed to memory.*
Note that the last two may be summed up in this way: the
sine of an angle is the cosine of its complement. Explain the
significance of co in cosine.
Some teachers prefer the words opposite and adjacent
instead ^perpendicular and base, but experience suggests that
for beginners the latter terms are preferable. The main
thing is to adopt one form of words and stick to it.
The secant, cosecant, and cotangent. These should be
remembered as the reciprocals of the cos, sin, and tan,
respectively. Give easy examples to show the appropriateness
of the forms beginning with co.
The ratios of common angles. The sin, cos, and tan of
the common angles 30, 45, and 60 should be memorized
Fig. 187
as soon as the nature of the three functions is understood.
Teach the boys to visualize the half square and the half
equilateral triangle the obvious aids to memory.
* Dp not despise some simple form of mnemonics when, with beginners, con
fusion is almost inevitable, as in the case of the three trigonometrical functions; e.g.
remember
(1) Tan =
(2) Sin ==
(3) Cos =
by the words Tanned Post Boy,
by the words Sign, Please, Henry,
by the words Costly Black Hat,
or some other form of catchy words,
PLANE TRIGONOMETRY
349
30
45
60
1
V3
sin
4
V2
T~
V3
1
cos
i
1
tan
1
V3
V3
A little later, the table should be extended to and to
90, and eventually to 180. When discussing the and
90 values, draw a series of rightangled triangles, beginning
with a very small acute angle A and very nearly 0, and ending
with an angle A very nearly 90. A discussion of just one
general figure, without reference to the actual values of
particular cases, is, with beginners, almost profitless. Do
not say that the tan of 90 is " infinity ", a term which is
beyond the comprehension of beginners. Adopt some such
noncommittal form of words as " immeasurably great ".
*
Fig. 188
The ratio for 15 is easily obtained from this figure,
derived from fig. 187. ABC is an isosceles rt. Z.d A, sides
1, 1, 2, angles 45, 45, 90. ABD is a half equil. A, sides
1, 2, v% angles 30, 60, 90. Thus ^CAD = 15, and
350 CRAFTSMANSHIP IN MATHEMATICS
CD = (\/3 1). From C, drop a perpendicular on AD.
Since CED is a half equil. A, CE = CD = ^ 3 ~ 1 .
pp A/3 _ 1
Hence sin 15 ' ~  From this the other ratios
of 15 are easily found, and then those of 75.
For 18, fig. 113fl is the key. The small angles of a regular
pentagram are 36, and hence the sine of half the angle is
  . Let the boys work this out for themselves; it is
4
a good exercise; the other ratios may be derived arithmetically,
but the first (the sine) must be established geometrically.
The derivation for multiples of 18 (36, 54, 72) is suitable
work a year later.
The following identities may readily be established
geometrically.
1. Sin 2 A + cos 2 A = 1. This is seen from a figure to
be a direct application of Pythagoras. Let the derivatives
also be noted: sin A = A/1 cos 2 A, cos A = A/1 sin 2 A.
2. 1 + tan 2 A = sec 2 A. Here a hint is necessary to the
boys to work " backwards ". We have to prove:
J? j BC 2 AB 2
i.e.
AC 2 AC 2 '
AC 2 + BC 2 AB 2
Fig. is 9 AC 2 AC 2 '
The boys now observe that the numerators form the simple
Pythagoras relation. Hence they write out:
AC 2 + BC 2 = AB 2 , (Pythag.)
. AC 2 BC 2 AB 2 . ,
'
/. 1 + tan 2 A = sec 2 A. Q.E.D.
The obvious derivatives should follow. Give several easy
PLANE TRIGONOMETRY 351
examples to verify the rule that if any one trigonometrical
ratio of an angle be given, the other ratios may all be cal
culated without reference to tables. But all fundamental
relations must be established geometrically. Geometry must
take precedence over algebra.
Heights and Distances
It is surprising what a great variety of problems, in
three as well as in two dimensions, may be solved by means
of the small amount of trigonometry already touched upon.
Give plenty of such problems until the sin, cos, and tan
are as familiar as the multiplication table, are, indeed, a
part of the multiplication table. Insist all along that every
problem on heights and distances is really a geometry problem
with an arithmetical tail, but that the arithmetic is made
easy for us because all the necessary multiplication sums
have been worked out and the answers put into a book of
tables, the multipliers having been given the rather fanciful
names of sin, cos, tan, &c. In every problem we are con
cerned with a triangle; the length of one side is always
given, and the multipliers in the book of tables enable us to
find the other sides; to find the multipliers, we have to know
the angles of the triangle. Fourfigure tables of natural
sines, cosines, and tangents, for whole degrees only, are
enough for beginners. Let logs wait. Let the problems be
easy and varied. Threedimen
sional problems may be in
cluded quite soon, though at
least a little solid geometry
should have been done pre W
viously.
When setting problems in
volving " bearings ", avoid, as ,.
a rule, the old terms " north Fig. 190
west ", " southeast ", &c., and
adopt the surveyor's plan, always placing N. or S. first,
352 CRAFTSMANSHIP IN MATHEMATICS
then so many degrees W. or E., thus N. 30 W., S. 60 E.
the angle always being measured from the N. S. line.
The drawing of figures for heights and distances. If a
figure lies wholly in a horizontal plane, there is seldom much
difficulty, especially if drawing to scale has been properly
taught in the Junior Forms. Figures in a vertical plane are
also readily drawn, though the angles of elevation and de
pression are sometimes confused by boys whose early practical
geometry has not been properly taught.
Consider this old problem:
From a point P in a horizontal
plane, an observer notes that a
distant inaccessible tower subtends
an angle of 30. He walks to
Q, a distance of 100/J., towards
the tower, and finds that the tower
then subtends 50. Find the height
Fig. 191 of the tower and the man's dis
tance from it.
Explain how easy it is to work with tangents, as the figure
readily shows.
(1) RS = PS tan 30, i.e. x = (y + 100) tan 30.
(2) RS = QS tan50, i.e. x=y tan50,
.'. (y + 100) tan 30  y tan 50.
Hence y can be found, then x by substitution. The long
succession of statements in some of the textbooks is un
necessary and merely serves to bewilder the boys.
The problem is, of course, easy enough. It is only when
the measured distance PQ is not in the same plane as PRS,
i.e. is not directly towards the tower, that the boys are baffled,
because of the difficulty of drawing a suitable figure in 3
dimensions.
We will deal with the threedimensional figure difficulty
in one or two problems:
A wall 12 ft. high runs east and west. The sun bears
PLANE TRIGONOMETRY
353
Calculate the breadth of the
S. 60 W. at an elevation of 32.
shadow of the wall on the ground.
This is taken from one of the best of our textbooks,
is, of course, very simple, yet S
I have given it to several lots of
boys, and the necessary figures
have nearly always puzzled them.
Had the sun been directly south,
a stick placed vertically at O
would have had its shadow cast
on the ground in the direction
ON (fig. 192). But as the sun was S. 60 W., the stick at O
would have had its shadow cast on the ground in the direc
It
W
N
Fig. 192
W
N
Fig. 193
tion OQ lt so that the shadow makes an angle of 30
the vertical plane (wall) in EW (fig. 193).
SUN
with
O
Fig. 194
But as the sun has an elevation of 32, the length of the
shadow of the stick RO would be Q 2 O, Q 2 being the far end
of the shadow on the ground (fig. 194). (During the day
(B291)
354
CRAFTSMANSHIP IN MATHEMATICS
this shadow would occupy a succession of positions, just as
if it were pivoted on the stick, following the sun round.)
We have to consider these two things, the direction and
the length of the shadow in a threedimensional figure. Let
ABCD be the wall running east west; it may be looked
upon as a close set of palings, with one paling RO taking
the place of the stick. Of course the shadow of the whole
wall will be cast, but we will first consider the shadow of
RO only. As the sun is at 32, the shadow must be cast
rS\JH
Fig. 195
somewhere on the ground as a length OQ. That " somewhere "
is given us by the sun's position (irrespective of its height)
at a particular time in the day, viz. S. 60 W., i.e. OQ will
make 30 with the eastwest wall, or ZBOQ = 30.
Now all the palings will cast shadows parallel to OQ,
and thus we shall have a belt of shadow, on the ground
BMNC, of a breadth equal to the perpendicular QP to the
wall. Thus we have to find the length of PQ.
To find the length of PQ we may solve the APQO in the
H.P. In the ARQO (vertical plane), RO = 12'; QO = RQ
cot32 = RO tan58 = 12' x 16 = 192'. In the APQO,
PQ = QO sin30 = 192' X 5 = 96'.
PLANE TRIGONOMETRY
355
For beginners a model is far better than a sketch; then
the angles do not mislead. Even a book held upright on the
desk to represent a vertical plane, and then a pencil placed
in position to represent a line in an oblique plane, will help
the eye greatly. But some long hatpins stuck vertically
into a board, with pieces of cotton tied round under the
heads (a snick made with the laboratory file will help to
secure the cotton), stretched and held fast by a twist under
the head of a drawingpin, will enable the boys to make in
a minute or two a model of almost any figure that may be
required.
Here is another problem and the provided figure from the
same excellent textbook. The figure has puzzled several
lots of boys.
" A hillside is a plane sloping at 27 to the horizontal. A
straight track runs up the hill at an angle of 34 with a line
of greatest slope. What angle does the track make with the
horizontal?"
" AB is the line of intersection of the hillside and a
horizontal plane ABC. AF, BE are lines of greatest slope
Fig. 196
meeting a horizontal at F, E. Let the track AD cut EF at
D. Draw DN, EC perpendicular to the H.P., ABC. Then
AN is the projection of AD on ABC. It is required to
find ZDAN = 0, say." Then follows the solution, simple
enough, of course, from considerations of the 3 it. Zd As
ECB, AND, AFD, the first two in V.P.s, the last in an oblique
356
CRAFTSMANSHIP IN MATHEMATICS
plane. A model with 3 hatpins at FM, DN, and EC, and
drawingpins at A, B, C, M, N, and connecting threads,
would make the whole
thing clear at once.
Otherwise a few shading
lines might be added, as
in fig. 197; the 3 planes
are then shown clearly.
Here is a simple
problem from another
book, the figure for which
has often given beginners
trouble. The extremity of the shadow of a flagstaff FG, 6'
high, standing on the top of a square pyramid 34' high, just
reaches the side of the base and is distant 56' and 8' respectively
from the extremities of that side. Find the sun's altitude.
Fig. 197
Fig. 198
FK = FG + GK = 6' + 34' = 40'. We have to find
the Z.FMK in the rt. Zd A FMK. In this A we know
FK; and we can find MK by Pythagoras from A KMN
in the plan (second figure):
KM = A/32 2 + 24 2 = 8 x 5 = 40.
TanFMK = $g = 1; /. ^FMK = 45.
PLANE TRIGONOMETRY
357
In practical problems, boys are constantly blundering
over compass bearings. Impress on the class that the difference
between the bearings of two distant objects is the angle made
by the two lines, drawn in the Jf/.P., from the observer to
the objects. If the objects are above the H.P., the difference
between the bearings is still an angle on the H.P., viz. the
angle between the two vertical planes drawn through the
observer and each of the objects. An observer is at S in a
H.P., his southnorth line being SN. PQ and RT are two
vertical poles. He takes the bearings of the two poles and
Fig. 199
finds that their horizontal angles are respectively N. 50 W.
and N. 55 E., i.e. the difference between their bearings is
105.
Now suppose he could not see the bottom of the poles,
because of an intervening hill. The observer would have
to point his telescope at the poletops P and R, and he could
then, if he wished, take the angles of elevation. But his pur
pose now is to take the difference between the bearings, and
he would therefore observe where each vertical plane con
taining the tilted telescope cut the horizontal plane. The
angle to be measured ( ZQST) would be exactly the same
as before. Impress on the boys that the observer could not
measure the angle PSR in the oblique plane; his theodolite
does not permit of that. And even if he could, the angle
would not represent the angle between the bearings.
Here is an illustrative problem. Find the distance between
the tops of the spires of two distant inaccessible churches. (It
would be taken rather later in the course.)
358
CRAFTSMANSHIP IN MATHEMATICS
Measure off a base line AB in a suitable position, and from
each end take the bearings of both spires, P and Q, draw
ELEVATION
(V P *o.tl> spines)
a ground plan, and mark in the Zs a, j8, y, 8. The
f may be calculated if wanted.
and
In the APAB, AB and the Zs are known;
.*. PA and PB can be calculated.
In the AQAB, AB and the Zs are known;
.'. BO can be calculated.
In the APBQ, PB and BQ are calculated, and ZS known,
/. PQ can be calculated.
Now examine the perspective sketch, with the 2 spires P'P
and Q'Q in position. We have to find P'Q'. We know AP,
BQ, PQ. Measure the /.s of elevation p and a\ P'P = AP
tan/), Q'Q = BQ tancr; hence P'P and Q'Q are known.
Hence in the elevation, everything is known except P'Q',
and this is easily calculated by Pythagoras.
Make the boys do this practically. Any two distant tall
objects will do.
PLANE TRIGONOMETRY 369
The Obtuse Angle
Angles up to 180 should be considered at an early stage,
but, before angles greater than 180 are considered, substantial
progress on the practical side of the subject is desirable.
Remind the boys that the rotating arm of the angle,
regarded as the hypotenuse of a rt. Zd A, may be carried
round from 90 to 180, the pivot being the point of inter
section of the coordinate axes. Refer to the work on graphs,
and the rule of signs for the second quadrant; all x values
measured to the left of the origin are regarded as negative.
If, for instance, we consider
the triangle BOC in the
second quadrant, the hypo
tenuse and perpendicular are
positive as in the first quad
rant, but the base OC is
negative. Proof? There is
none. It is merely an accepted Fig. 201
convention.
Suppose, then, we have an angle greater than 90, say
145. How am I to find the value of its sine, cosine, and
tangent?
Exactly as before. From any point on the rotating
arm OB, drop a perpendicular to the fixed arm OA (pro
duced backwards, because necessary), and take the ratios
in the same way as we did for acute angles. But remember
the signs. For this angle 145, the hypotenuse is OB, the
base OC, the perpendicular BC.
TT IJfO BC 1JKO OC 1 AtaO BC
Hence: sin 145 = ; cos 145 = , tan 145 =
OB' ~" w OB~
It may not look as if the perpendicular BC concerned the
angle AOB (145), but how else could a perpendicular for
145 be drawn?
Now consider an acute angle equal to the angle BOC
in fig. 201. Evidently it is (180  145) or 35. Fig. 202
360
CRAFTSMANSHIP IN MATHEMATICS
shows OB' equal to OB in fig. 201. Hence the triangle
B'OC' has sides of exactly the same length as the triangle
TVP' OP' '
BOC. SinS5' = fL; cos35 = ; tan35 =
Fig. 202
Comparing the ratios for 145 and 35, we see that:
sin35 = sinl45,
cos35 = _cos!45,
tan 35 tan 145.
The same thing must apply to any pair of angles whose sum
is 180. Thus we may say,
sinA = sin(180  A),
cosA = cos(180 A),
tanA = tan(180  A).
If the above demonstration is attempted from a single
figure (as it might well be from fig. 201), slower boys will
inevitably be confused.
It is an excellent plan to make boys in Upper Sets express
their ratios in terms of coordinates, i.e. to call the rotating
arm r, and its extremity P (#, y).
Give plenty of oral practice in the obtuse angle relations,
e.g. tanlOO = tan(180  100) = tan80 = 567.
PLANE TRIGONOMETRY 361
The General Triangle and its Subsidiary
Problems
Before proceeding to the solution of triangles, revise
carefully the geometry of congruent triangles, and note
what various sets of data are necessary and sufficient for
copying a triangle. A triangle is determined uniquely if we
are given (1) the 3 sides, (2) 2 sides and the included angle,
(3) 1 side and 2 angles. If we are given 2 sides and the angle
opposite one of them, there may be 2 solutions, or 1 solution,
or no solution.
A triangle cannot be determined unless the data include
at least one side.
Thus the necessary data include 3 elements, at least
one of them being a length.
All the formulae in this section must be established
geometrically. As geometrical exercises they are all first
rate.
1. In any triangle ABC, a = b cosC + c cosB. Show
that this relation holds good for both acuteangled and obtuse
angled triangles. It is simply a question of dropping a per
pendicular and considering separately the two resulting
rightangled triangles.
Do not forget the sister expressions in this and sub
sequent formulae. The one thing to keep in mind is the cyclic
order of the letters A, B, C, and a, A, c. For instance, the
above identity may be written,
b = c cos A + a cosC,
or, c = a cosB + b cos A.
2. The sine formula. In any triangle ABC,
sin A sinB
As before, show that the relation holds good for
sinC
obtuseangled as well as for acuteangled triangles.
362 CRAFTSMANSHIP IN MATHEMATICS
The " ambiguous case " should receive special attention.
Link up the work with the closely analogous case in geometry.
In fact the problems are the same. Readily understood as
they generally are, they are often half forgotten. They must
be regarded as sufficiently tedious and troublesome as to
merit special and repeated attention.
3. The cosine formula. In any triangle ABC, c* = a 2 f ^ 2
~2ab cosC. Again be careful to consider both acuteangled
and obtuseangled triangles. Link up carefully with Py
thagoras and its extensions (Euclid, I, 47; II, 12, 13). The
solution is straightforward and seldom gives trouble.
When solving triangles, use sine or cosine formula?
If given (1) 3 sides,
or (2) 2 sides and in,
eluded angle,
use cosine formula for first
operation; then continue
with the quicker sine for
mula, using it to find the
smaller of the two remain
ing angles.
If given (3) 2 angles and 1 side, } use sine
or (4) 2 sides and a notincluded angle,) formula.
N.B. (1) If given 3 sides, find the smallest angle first.
(2) If the given triangle is isosceles, use neither
formula, but drop a perpendicular to the
base.
*
4. The tangent formula. In any triangle ABC (where
b>c),
tanj(B C) _ b c
tan(B + C) ~~ b + c
This is a useful alternative, more suitable for log calculations,
when 2 sides and the included angle are given. The cos
formula is often cumbrous in application, not being suitable
for log calculation.
PLANE TRIGONOMETRY 363
The boys must learn to establish the formula geometrically,
from first principles, and not derive it from other trigono
metrical formulae. But for beginners it is generally puzzling.
Begin by giving them a particular case to which they may
apply the formula. Let the sides of a triangle be, say, 11,
13, 16, and let the boys work out the angles from their cos
B a 16 O
Fig. 203
and sine rules, using fourfigure tables. The angles shown
in the figure are, to the nearest minute,
tan(B  C) = tan (53 47' 43 3')
tanJ(B + C) tan(53 47' + 43 3')
tan522 ; 094 1
Again:
tan 48 25' 1127 12*
bc 1311 1
b + c 13 + 11 12
Thus they see that, at least in this particular case, the theorem
holds good. Working out a particular case in this way, they
grasp the fact that tan(B C) is, after all, just the tan of
a simple angle. So with (B + C).
The problem now is to devise a figure which shall
actually show these angles (B C) and f (B + C); also
the sum (b + c) and the difference (b c) of the sides.
There are two subsidiary points to note first.
364 CRAFTSMANSHIP IN MATHEMATICS
(1) In any triangle,
since A + B + C = 180,
or
Again:
or
Similarly:
+
_
90 .
T'
B f C
BC
22
t_A _ A = C
Give plenty of oral work on these points, with blackboard
figures to illustrate.
(2) How have we been able in geometry to show the
sum and difference of two sides of a triangle?
Fig. 204
The sum of b and c may be shown by swinging round AB
on A to AE, so that AE = AB; hence CE = (b + r); the
difference may be shown by cutting from AC a part AD equal
PLANE TRIGONOMETRY
365
to AB; thus DC = AC AB = (b c). The same figure
shows (B C). For (fig. ii), since m n + jf>, .". m' = n + />>
/. m' + p = n + 2p, or B = C + 2p; ;. == J(B C).
(i) Now we may draw the required figure.
With centre A and radius AB, describe the circle EBD,
and produce CA to E. Evidently EC = (b + c), DC = (b  c).
Join EB. Z.E (at circf.) = ^A (at centre). We know that
/.DEC (B C), but there seems to be no obviously
Fig. 205
simple way of using it. But if we draw CF parallel to BD,
to meet EB produced in F, Z.BCF = ZDBC = (B C).
Again, in the rightangled triangle EFC, since /.E = A,
ZECF = 1(B + C).
Thus we have the two angles and the two lengths for the
tan formula:
BF
tan^(BC) = tanBCF = FC = BF = DC _ b c Q E D
tanJ(B + C)~~tanECF EF EF EC b + c * *
FC
(ii) A boy might very well ask if we could use the figures
366 CRAFTSMANSHIP IN MATHEMATICS
made by drawing the circle with radius AC instead of AB.
Exterior ZA at centre = B + C; /. ZD at circum
ference = ( B + C).
Fig. 206
Also /.BCD = ZB LD (ext. L property) = ZB
 (B + C)  KB  C).
We may take the tan of the last angle by dropping the
BF.
BF
tan(B~C) _ tanBCF FC _, FD _ BD _ bc QED
tan (B + C) tanBDF BF FC BE b + c ' "
FD
There is no essential difference between the two proofs.
(iii) Or a boy might ask if we could not derive the angle
+ C) from the JA obtained by actually bisecting A.
PLANE TRIGONOMETRY
367
Let AD be the bisector, and let CD meet it at right angles.
Draw BF perpendicular to CD produced, and BE perpen
dicular to AD.
Evidently ZBCF
= KB C), and
Z.AEE = ZACD =
KB + C).
The figure does
not give us a length
AC + AB (= b + c),
or a length AC AB
(= b c). But we
can project AB and Fig 20?
AC on to FC; FD
(= BE) is the projection of AB, and DC is the projection
of AC, and so we may obtain what we want in this way:
(1) AD = b sinKB + C); AE == c sin(B + C);
.. BF = AD  AE
Vi(Bc)
(2) DC = b cosKB + C); FD = c cos(B + C):
/. FC = DC + FD
= (b + c) cos(B + C).
. BF _ (b  c) sinj(B + C) _ b  c
(6) FC ~ (4 + c) cos KB + C) V+c a i(
But
BF
= tani(B
C);
.
b
tan (B + C) = tan KB  C),
b c = tan J(B C)
b + c ~~ tan KB + Cj"
or
This last method is not quite so simple as the first, but it
appeals to A Sets to whom alone (perhaps) it should be given,
368 CRAFTSMANSHIP IN MATHEMATICS
5. Other formulae that should be worked out geometrically:
(i) Cos2a = 2cos 2 a 1 = 1 2 sin 2 a.
(ii) Sin2a = 2 sin a cos a.
02 I 2 _ C 2
(iii) CosC     (and thence, algebraically,
. 
the halfangle formula, sinA == y ~ *) (* ~ c )
o \ be *
&c.).
(iv) Area = \bc sin A; &c.
(v) Circles of a triangle: circumscribed, inscribed,
escribed.
(vi) Medians, angle bisectors, pedal triangle, ortho
centre, &c.
The geometry of these basic formulae is the important thing.
The derivatives may be obtained algebraically.
Angles up to 360. The Four Quadrants
It is best to begin by showing the boys how surveyors
in their work often find it an advantage to consider angles
up to 360. We have therefore to decide how the ratios of
angles between 180 and 360 can be expressed. Thus we
have to consider the 3rd and 4th quadrants.
Remind the boys that there are no proofs of our conven
tions concerning the signs in the four quadrants. The con
ventions are just a matter of convenience, arrived at by
general consent, and consistent with one another. It is this
consistency which is the important thing. The boys must
be drilled in the quadrant signs until the last shred of doubt
disappears.
44 Plus: right and above,"
44 Minus: left and below."
Fig. 308
PLANE TRIGONOMETRY
369
Other important memos.
1. The fixed arm of the angle is always in the 3 o'clock
position.
2. The rotating arm of the angle always moves counter
clockwise.
3. Never take a short cut by moving clockwise.
From any point in the rotating arm we may drop a per
pendi cular PM on the abscissa, form a rightangled triangle,
M.
(0
(ii)
Fig. 209
and so take any ratio of any angle in any quadrant. Taking
e.g., the tangent, we have:
OM 2
tanXOP, = d^ =
tanXOP =
ta 4
~
+OM 4
OM 2 '
OM'
.
OM 4
(1291)
25
370
CRAFTSMANSHIP IN MATHEMATICS
Beginners are often puzzled about the reentrant angles in
the 3rd and 4th quadrants. Make them understand that if
they take the smaller angles in these quadrants, they have
taken a clockwise rotation of the moving arm, and this is
not allowable. (Postpone the consideration of negative rota
tions until the main principle is grasped thoroughly.)
As already suggested, an alternative plan is to call the
length of the rotating arm r, and to call the point P which
we fix in it (x, y), x and y being the coordinates of the point.
But if the boys are at first well drilled in the use of the terms
hypotenuse, base, and perpendicular, these terms will probably
continue to be used, at least mentally. In A Sets, the co
ordinate notation is preferable: its advantages are obvious.
Make the boys memorize the following scheme: it merely
amplifies what was said on a previous page.
i.
2.
3.
4.
Sin . .
+
+
Cos ..
f
+
Tan . .
+
+
" Sin, cos, and tan are + i n the 1st quadrant, and each is + in
one other, viz. sin in 2nd, tan in 3rd, cos in 4th."*
Give plenty of oral work on the ratios of angles in all
four quadrants. Boys should recognize the landmarks 90,
180, 270, 360, and know at once in which quadrant a
given angle occurs.
Beginners are often caught: they take the complement
of the angle instead of the angle itself. Point out again and
again that whatever angle we may have in the first quadrant
there must be angles with exactly the same numerical ratio
in the other three quadrants. The four resulting triangles
* One or two schools use this mnemonic:
s *  all
T I
" positively all silver tea cups ".
PLANE TRIGONOMETRY
371
formed by dropping a perpendicular from the same point
P on the rotating arm must be congruent.
/ \
N^
1
+ 128 +128
>
"*"
X
as:
232
(iii)
Fig. 210
Note the 4 angles: 52; 180  52 = 128; 180 + 52
= 232; 360  52 = 308.
1*28
Note also the 4 tangents:* tan52 = +  ; tan!28
=  ; tan232 =
; tan308 = 
The angle in the second quadrant is obtained by sub
tracting 52 from 180. 
The angle in the third quadrant is obtained by adding
52 to 180.
The angle in the fourth quadrant is obtained by sub
tracting 52 from 360.
Quite by chance the angle in the second quadrant (128) has the appearance of
being 100 times the value of the tangent (128).
372
CRAFTSMANSHIP IN MATHEMATICS
The four angles do not form an arithmetical progression,
and they cannot do so unless the angle in the first quadrant
is 45.
Any such group of 4 angles forms a symmetrical figure:
Whenever we take a trigonometrical ratio of an angle
from the tables, the angle is one belonging to the first quad
rant. But there are three other angles having the same
numerical value. If a is the angle in the first quadrant, the
other three are 180  a, 180 + a, 360 a. But each
ratio in each quadrant has its own signs as we have already
seen.
To evaluate the ratios of angles greater than 90, we may
remember the formula, 180 a, though this is really
intended to include angles greater than 360. Let the boys
make up this general formula from an examination of a
number of particular cases.
First Notions of Periodicity
The boys are already familiar with the notion that the
rotating arm of the angle may proceed beyond one revolution;
the movement of the pedal of an ordinary bicycle serves to
convey the notion of angles of w360 or w360 + a. Show
clearly that the ratios of any angle a are exactly the same as
those of any angle that differs from a by any number of
PLANE TRIGONOMETRY
373
complete revolutions. Thus, sin(w360 + a) sin a, where
n is any integer; so with all the ratios. For example,
cos 700 = cos (720  20)  cos (2 . 360 20)
= cos 20 = cos 20.
Give ample oral practice to emphasize the fact that the addition
or subtraction of any multiple of 360 does not alter the value
of any ratio of an angle. The general rule may be expressed:
" If a is an angle, any ratio of %nir i a is numerically equal
to the same ratio of a ". The sign to be attached depends
on the quadrant. (The radian notation should be familiar
by this time.)
For purposes of illustrating continuous functions, graphs
may be obtained, with sufficient accuracy, by 30 and 60
360
360
Fig. 212
parallels and perpendiculars as in fig. 183. The two inter
mediate points in each quadrant are enough to determine
the curve fairly readily. The boys should be able to sketch
the curves rapidly and should become thoroughly familiar
with them. They should note that if the graph of cos a be
moved 90 units along the x axis, it coincides with that of
sin a, and that this is equivalent to saying that sin (a + 90)
= cos a.* Superpose the cos graph on the sine graph and
discuss the intersecting points and the ratios of the angles
indicated by those points. Draw a continuous sine graph up
to 5?r or GTT. Select some first quadrant angle, say 40,
raise a perpendicular to cut the graph, and through the
point of intersection run a parallel to the x axis and another
the same distance below the axis. Discuss and compare the
Slower boys will confuse 90 f A with 180 A.
necessarily different unless A = 45.
Show that they are
374
CRAFTSMANSHIP IN MATHEMATICS
sines of all the angles indicated by the successive points
of intersection. Show clearly that there is a period of 27r,
and that sin# may therefore suitably be called a periodic
function of x. So with cos x. Tan x is likewise a periodic
7
~3ir
\
lig. 213
function, but with a period of TT (not 2??); show how this
may be inferred from the parallel tan curves.
Compound Angles
1. Sin(A + B) = sinA cosB + cosA sinB.
2. Cos(A + B) = cosA cosB sinA sinB.
3. Sin(A B) = sinA cosB cos A sinB.
4. Cos(A B) cos A cosB + sinA sinB.
Beginners naturally think that sin 50 = sin 20 + sin 30,
that cos 70 = cos80 cos 10. Give a few examples, with
free reference to the fourfigure tables, to show that this is
not so.
Of the four identities named above, at least one should
be proved geometrically and mastered thoroughly. The
neatest method is the projection method, and with A Sets
the general case can readily be proved by this method. With
B Sets and certainly with C Sets the problem is best con
sidered merely from the point of view of positive acute
angles. All the books give the solution, but the boys should
be taught to analyse the conditions of the problem, not
merely to follow out a book solution.
The following sequence of arguments is suitable for teach
ing purposes.
PLANE TRIGONOMETRY
375
Let OX rotate through ZA to OC, then through ZB
to OD; in its complete journey to OD it has rotated through
the complete L (A + B). We have to prove that sin (A + B)
= sin A cos B + cos A sin B, and in connexion with the
three angles this means the consideration of five ratios, viz.
Fig. 214
the sines of A, B, and A + B, and the cosines of A and B.
We will try to show all these in one figure.
Evidently we require three perpendiculars, since there
are three angles.
(1) From any point P in OD, drop the JL PN to OX.
The sine of L(A + B) may be considered from the rt. Zd A
PON.
(2) From P, drop a L PQ on OC. The sine and the cos of
ZB may be considered from the rt. Zd A POQ.
(3) From Q, drop a L QR on OX. The sine and cos of
ZA may be considered from the rt. Zd A QOR.
When we have to prove that a simple expression is equal
to a more complex expression, it is a good general rule to
begin with the latter, try to simplify it, and get back to the
former. Thus we may begin:
sin A cosB f cos A sinB
= cm QO OR PQ
QO ' OP + QO ' OP'
But how are we to proceed now? True the OQ's seem to
376
CRAFTSMANSHIP IN MATHEMATICS
cancel out in the lefthand term, but we do not seem to be
able to simplify any further.
Since a circle will go round ONQP (on OP as diameter),
Z.MPQ = ZA. If then we draw QM PN, we have a
N R X
Fig. 215
APMQ similar to AQRO. Thus, as far as ratios are con
cerned we may consider APMQ instead of AQRO. Now
let us try simplification again:
sin A cosB f cos A sinB
Q0 , PM PQ
'OP PQ'OP
QR . PM PN . /A , m
c=  +  =  = sin (A + B).
OP OP OP V J
We may now set our proof as an examiner would expect to
see it.
QR PM
_ QR OQ PM PQ
"~OP'OQ OP 'PQ
(each term multiplied by 1)
= QR OQ PM PQ
OQ ' OP + PQ ' OP
= sinA sinB + cosA sinB. Q.E.D.
PLANE TRIGONOMETRY
377
The three analogues now follow on simply. All four identities
should be verified by a few particular cases (4figure logs will
do), e.g.
(sin55 cos 25 + cos 55 sin 25,
sin 80 =
IsinlO cos 70 + cos 70 sin 10.
Fig. 216
Some teachers prefer this proof instead:
Let the acute Z.s A and B
be the Zs of a A ABC. Draw
a circle round the A, and the
radii OA, OB, OC. Evidently
ZAOC  2B, ZBOC = 2A.
If Js from the centre be
drawn, they bisect the sides
of the A . Hence, AB = d sin
(A + B), AC rf sinB, CB
= rfsinA; also AB = AC
cos A + CB cosB. By equat
ing the first and last of
these, and substituting from the 2nd and 3rd:
d sin(A + B) = AC cos A + CB cosB
= d sinB cos A + d sin A cosB,
or sin(A + B) sinA cosB + cosA sinB.
This proof does not seem to appeal to boys so readily as the
former does.
We now come to the general case. B and C Sets find it
difficult, and as a rule it should be given to A Sets only.
Of the various methods of proof the two following are the
simplest for teaching purposes.
1. The Projection Method. This method is productive
of mistakes unless the boys have mastered the elementary
principles of projection.
Give the class one or two preliminary exercises of the
following kind:
378
CRAFTSMANSHIP IN MATHEMATICS
The angle A of the regular pentagon ABODE touches the
X axis, with which AB makes
an angle of 12. Find (1) the
horizontal distance of the ver
tex D from A, and (2) the
height of D above the X axis.
(1) Horizontal distance of
D from A = projection of
AB + projection of BC +
projection of CD. Remem
ber that projection means
projection with proper sign
attached, and we must take
the Zs which AB, BC, CD
make with the + direction of OX. Take AB as unity.
/. Distance AH  AB cos 12 + BC cos 84 + CD cos 156
= 9781" + 1045" 9135"
= 1691".
(2) Height of D above OX.
Height HD  OM + MN + NP
= AB sin!2 + BC sin84 + CD sin!56
= 2079" + 994:5" + 4067"
= 16091".
Now we come to the identity sin(A + B) = sin A cosB + cos A
sinB. Let the Zs be the same as in fig. 214. From any
Fig. 218
PLANE TRIGONOMETRY
379
point P in OD, draw PQ L OC. Project the three sides of
the APOQ on the Y axis.
OM = ON + NM,
.'. projection of OP = sum of projections of OQ and QP.
/. OP sinXOP  OQ sinXOC + QP cosXOC.
.'. OP sin(A + B) OP cosB sinA + OP sinB cosA,
(OQ = OP cosB, QP = OP sinB)
i.e. sin (A + B) = sin A cosB + sinB cos A.
In a similar manner, by projecting the three sides on the
X axis, we may prove that
cos (A + B) = cos A cosB sin A sinB.
Note that the method is perfectly general, being applicable
to any angles.
2. The Cosine Rule Method. This is based on the
rules (1) that cos A = + c ~~ a _ and ^ 2 ) that if P and Q
2bc
are the two coordinate points (x l9 jy t ), (x 2 , j> 2 ), then PQ 2
The identity usually considered is cos (A B) = cos A
cosB + sin A sinB, the
others being treated as
derivatives.
Whatever two angles
are given, the initial line
for each is the positive
direction of the X axis.
Note that we are taking
the difference between
two angles, not their
sum. No matter what (CosA,s^A)
two angles are taken, Fig. 219
cosPOQ = cos(A B).
The simplest way is to take a circle of unit radius, and
Q(COS
380 CRAFTSMANSHIP IN MATHEMATICS
to let the coordinates of P be (cos A, sin A) and of Q, (cosB,
sinB). Note that OP = OQ = 1; hence the denominators
of the cosine ratios need not be written.
Cos (A B)
= cosPOQ
= Qp2 + Q 2  PQ 2
2OP.OQ
_ 2  PQ 2
2
2  ((cos A  cosB) 2 + (sinAsinB) 2 }
.
2cosAcosB + sin 2 A + sin 2 B 2sinAsinB}
2
2 {2 (2 cos A cos B + 2 sin A sinB) }
2
= cos A cosB + sin A sinB.
There does not seem to be much to choose between this
method and the project! ve method. To able boys both
methods appeal. To boys of poor mathematical ability,
both methods are equally hateful.
One or other of the four identities, preferably the one
proved by the projective method, should be regarded as
basic, and the others should be treated as derivatives.
Other necessary derivatives are:
(1) 2 sin A cos B = sin (A + B) + sin (A B), and its
three analogues.
C + D C D
(2) sinC + sinD = 2 sin cos  , and its three
analogues.
(3) tan(A + B) = tanA + tanB
v ' v i y
/A\ OA
(4) tan 2 A =  .
v ' 1 tan 2 A
(5) sin3A = 3 sin A 4 sin 3 A.
(6) cosSA = 3 cosA + 4 cos 8 A.
PLANE TRIGONOMETRY 381
All the formulae should be learnt off. If mnemonics can be
devised, they will help the lame ducks much.
Let the boys verify all the formulae established, by means
of a few simple exercises. Use fourfigure logs for this purpose,
and so cover a good deal of ground in a short time. For
instance, show that
2 sin 50 cos 24 = sin (50 + 24) + sin (50 24)
= sin74 + sin26,
(2 X 7660 X 9135) = (9613 + 4384), &c.
Books to consult:
1. Trigonometry, Siddons and Hughes.
2. Elementary Trigonometry, Durell and Wright.
3. Advanced Trigonometry, Durell and Robson.
4. The Teaching of Algebra, Nunn.
5. Elementary Trigonometry, Heath.
6. Trigonometry, Lachlan and Fletcher.
7. A Treatise on Plane Trigonometry, Hobson.
CHAPTER XXVII
Spherical Trigonometry
Spherical trigonometry enters into the work of the map
maker, the navigator, and the astronomer; also into the
work of the surveyor if that work extends over larger areas,
as in the case of the Ordnance Survey. But for an under
standing of the essentials of surveying, mapmaking, navigation,
and astronomy, little more than the A, B, C of spherical
trigonometry is required, and all this can be included in a
very few lessons. The elementary geometry of the sphere
should already have been done.
The following are the chief points for inclusion in the
necessary elementary course. (Many of the difficulties can
382 CRAFTSMANSHIP IN MATHEMATICS
be elucidated by the use of simple illustrations. The orange,
with its natural sections, is very useful. Wellshaped apples
lend themselves to the making of useful sections. A slated
sphere, mounted, should always be available).
1. Great and small circles.
2. Shortest distance that can be traced between two
points on the surface of a sphere the arc of the great circle
passing through them. (A simple experimental verification
is good enough for beginners.)
A suitable argument: If a string be stretched between
two points on the surface of a sphere, it will evidently be
the shortest distance that can be traced on the surface between
the points, since, by pulling the ends of the string, its length
between the points will be shortened as much as the surface
will permit. Any part of the stretched string, being acted
on by two terminal tensions, and by the reaction of the
surface which is everywhere normal to it, must lie in a plane
containing the normal to the surface. Hence the plane of
the string contains the normals to the surface at all points
of its length, i.e. the string lies in a great circle. (Sixth
Form boys ought to appreciate such an argument.)
3. Axes; pole and polar.
4. Primary and secondary circles.
5. The angle between two great circles is measured by:
(i) the angle between their planes,
(ii) the arc intercepted by them on the great circle
to which they are secondaries,
(iii) the angular distances between their poles.
6. The spherical triangle that portion of the surface of a
sphere bounded by the arcs of three great circles. Parts:
3 sides and 3 angles.
7. Since 3 great circles intersect one another to form 8
triangles, that particular triangle is selected which has 2, or
if possible 3, sides each less than a quadrant.
Cut an orange or an apple into two equal parts;
SPHERICAL TRIGONOMETRY 383
hold the two parts together, and cut again into two
equal parts, this time by a plane oblique to the first;
hold the four parts together, and cut still again into
two equal parts, by a plane oblique to both of the
other planes.
8. The analogy between theorems in plane and spherical
trigonometry, e.g. any two sides of a triangle are together
greater than the third.
9. Polar triangles, i.e. triangles so related that the vertices
of the one are the poles of the sides of the other.
10. Angular limits of the sides and angles of a spherical
triangle.
11. Fundamental formulae:
(i) Any spherical triangle:
. cos a cosi cose , , 7 x
cos A = :  . (and analogues).
sm0 sine
(ii) Rightangled triangles:
sin A sin a/sine;
cosA = tani/tanc;
tan A = tanfl/sini.
(iii) Sine rule:
sin A sinB sinC 2n
sin a sin b sine sin a sin b sine*
All the proofs are simple. The only trouble is in the drawing
of suitable figures.
12. The Latitude problem. This is perhaps the most
important of the elementary problems of the sphere.
The navigator's " dead reckoning " depends on his know
ledge of two things: (1) his course (direction), (2) the distance
run (determined by log). He has to resolve his distance
course into separate mileage components of northing and
southing, easting and westing. Then he has to convert his
384 CRAFTSMANSHIP IN MATHEMATICS
northing and southing mileage into degrees and minutes of
latitude, his easting and westing into degrees and minutes of
longitude.
There is no difficulty with the former. The meridians
of longitude are all great circles. When we know the length
of the circumference of these circles, a simple calculation
will give the change of latitude produced by a given northing
and southing. (Polar circumference 24,856 miles; therefore
length of degree of latitude = 69 miles;  6 1 6 of 69 miles
= nautical or sea mile = 6080 ft. Thus 60 sea miles 1
degree of latitude, and 1 sea mile = 1 minute of latitude.
" Knots " = sea miles per hour.)
But parallels of latitude are small circles decreasing from
the equator to the poles. Only along the equator itself does
1 sea mile imply 1 degree of longitude. We have to discover
a law which the length of a degree of
longitude follows.
This law does not show a length
proportional to the distance from the
pole. The greatest distance between
two meridians is not halved at 45,
but at 60. Why has the parallel of
60 half the circumference of the
equator?
Fig. 220 CE = radius; A = point in lat.
60. Let figure rotate on PP'. The
circle will trace out the surface of the globe, E will trace
out the equator, and A the parallel of 60 of which AB
is the radius.
CE  CA = R (say).
Then AB = AC sin ACB,
= Rcos60,
= *R.
Since AB = R, circf. of the 60 parallel = \ length of
equator, .*. the length of a degree in 60 lat. is half the length
of a degree along the equator.
Thus a voyage of a given number of seamiles along the
SPHERICAL TRIGONOMETRY
385
60th parallel implies a change of longitude twice as great
as the same distance along the equator. With the help of
the slated globe, show the class how short the degrees of
longitude necessarily are in the neighbourhood of the Pole.
We give a suitable figure* for showing the general case.
If R be the radius of the equator, and r the radius of the
parallel of latitude A, passing through a given point, then
r = R cosA.
Q = Lat. 0, long. 0.
P' = Lat. X, Long. 0.
V = Lat. X, Long. P'V west.
T = Lat 0, Long. QT west.
Difference of longitude of M and V = arc MV = Z.MKV = LI.
Give the boys the little problem to prove that the length
of 1 minute of longitude measured along a parallel of lati
tude A is, 1 nautical mile X cosX.
* The figure is designed to show merely the main geometrical facts. When the
boys are familiar with these facts, the correct notation of polar coordinates, and the
accepted astronomical sign convention, should be introduced.
(E291) 26
386
CRAFTSMANSHIP IN MATHEMATICS
It requires very little skill in soldering to make a wire
model, and then the demonstration is exceedingly simple.
In spherical geometry and trigonometry, good figures are
essential, or very few boys will understand the problems
considered. Here is an example of a problem from one of
our very best books on the subject. We reproduce the original
figure.
The excess of the sum of the
three angles of a spherical triangle
over two right angles is a measure
of its area.
Let ABC be a spherical tri
angle; then, since the sum of the
three spherical segments (lunes)
ABA'C, A'BC'B', ACBC', ex
ceeds the hemisphere ACA' by
the two triangles ABC, A'B'C';
and since,
(i) the measures of the three
spherical segments are, respec
tively, the angles A, B, C, of the spherical triangle,
(ii) the measure of the hemisphere is 2 right angles,
/. the sum of the three angles exceeds 2 right angles. . (i)
If A is the number of degrees in the angle A, S the surface
A
of the hemisphere, the area of the spherical segment . S;
180
.'. since ABC is equal to its symmetric triangle A'B'C', the
result of (i) is that if S is the area of the spherical triangle,
/A + B + C
I 180
= 2S,
or
L = A + B + C  180
360 ' '
i.e. the area S is proportional to the excess of A + B + C
over 2 rt. Zs.
SPHERICAL TRIGONOMETRY
387
I have given this theorem to boys on several occasions,
but they have almost invariably failed to visualize the figure
properly. They failed to pick out the spherical segments.
We append four new figures. The first shows the spherical
triangle plainly; the next three show the three lunes, separately
shaded. The real trouble is that half the second lune (iii),
viz. the part A'B'C' (= the symmetric triangle of ABC) is
not visible. When the shaded lunes of ii, iii, iv are added
together, it is seen that the A ABC is included twice and
the hidden A'B'C' once. Hence the sum of the three shaded
areas exceeds the hemisphere by the two triangles ABC
and A'B'C'.
Books to consult:
1. Spherical Trigonometry, Murray.
2. Practical Surveying and Elementary Geodesy, Adams.
CHAPTER XXVIII
Towards De Moivre. Imaginaries
Interpretation of V 1
" Please sir, what is the good of De Moivre's theorem?
What is it really all about? What is the use of talking about
imaginary roots to equations?"
388 CRAFTSMANSHIP IN MATHEMATICS
Thoughtful boys often ask such questions. It is our business
to see that our answers satisfy them.
The symbol V 1, if interpreted as a number, has no
meaning. But algebraic transformations which involve the
use of complex quantities of the form a + bi (where a and
b are numbers, and /= V 1) yield propositions which do
relate purely to numbers, and those propositions are now
known to be rational and acceptable
Boys should understand that algebra does not depend on
arithmetic for the validity of its laws of transformation. If
there were such a dependence, it is obvious that as soon
as algebraic expressions are arithmetically unintelligible, all
laws respecting them lose their validity. But the laws of
algebra, though suggested by arithmetic, do not depend on
it. The laws regulating the manipulation of algebraic symbols
are identical with those of arithmetic, and it therefore follows
that no algebraic theorem can ever contradict any result
which could be arrived at by arithmetic, for the reasoning
in both cases merely applies the same general laws to different
classes of things. If an algebraic theorem is interpretable
in arithmetic, the corresponding arithmetical theorem is
therefore true. Sixth Form boys seem to gain confidence
when once they realize that algebra may be conceived as
an independent science dealing with the relations of certain
marks conditioned by the observance of certain conventional
laws.
It is true that the presentday use of imaginary quantities,
in accordance with the authoritative interpretation now
given them, does not involve any sort of contradiction and is
therefore presumably valid, for absence of logical contradic
tion is certainly a good test of valid reasoning. But Mr.
Bertrand Russell is perhaps going a little far when he
says (Prin. of Maths., Vol. I, p. 376) that the theory of im
aginaries has now lost its philosophical importance by ceasing
to be controversial. There is still a hesitancy in the treat
ment of the subject in Sixth Forms, which suggests that in
TOWARDS DE MOIVRE. IMAGINARIES 389
the minds of at least some teachers there is a lingering doubt
about the accepted interpretation.
Let the early treatment of the subject be frankly dogmatic.
Let discussions as to validity stand over for a while.
Define the symbol V 1 merely as an expression, (1)
the square of which = 1, and (2) which follows the ordinary
laws of algebra. And deduce the inference that since the
squares of all numbers, whether positive or negative, are
always positive, it follows that V 1 cannot represent any
numerical quantity.
Deduce the further inference that, since V a 2 V 1 X a 2
== V 1 X #, V a* cannot represent any numerical quan
tity. Thus V I X a may be called an " imaginary "
expression. It therefore follows that such a statement as
A + BV 1 = a + bV 1 can only be true when A = a
and B = b. _
Numbers like a + bV 1, where a and b are real numbers,
which consist of a real number and an imaginary number
added together, are called complex numbers.
At this stage it is advisable to revert to the significance
of ordinary negative quantities. If +a indicates a certain
number of linear units in some chosen direction, a indicates
the same number of linear units in the same line but in the
opposite direction. Hence when working out, with algebraic
symbols, a problem concerning distance, we interpret the
minus symbol to mean a complete reversal of direction.
It is desirable to take some little trouble to convince the
pupils that, on the face of things, there is nothing in the
expression a + b\/l to make it more " absurd " than in
an expression like x. The result symbolized by b a
where b is less than a is certainly " imaginary ", unless we
add to the conception of magnitude, which necessarily
belongs to it as a number, the further conception of direction.
Quantities which contain V 1 as a factor are obviously
in some ways very different from quantities which do not
contain it.
390 CRAFTSMANSHIP IN MATHEMATICS
What interpretation, then, can be given to the result of
multiplying a distance by V 1? Argand put forward an
ingenious hypothesis, which has now received general
acceptance.
As we have seen, the effect of multiplying a distance by
1 is to turn the distance through two right angles.
Hence, whatever interpretation we give to V 1, it
must be such that the multiplication of a distance by
V^ X V^l,
i.e. by 1, must have the effect of turning a distance through
two right angles.
Thus it seems worth while to consider how far we may
interpret the effect of multiplying a distance by V 1, by
supposing that it turns the distance
through one right angle. Evidently
we have to devise some scheme by
which a reversal of direction will
be effected in two identical opera
tions.
One possible plan is to revolve
OH through a right angle either in
the direction of S or in the direc
tion of T, for each of these opera
tions, if repeated, would bring H
into coincidence with K. Further double applications of
the same operation would successively bring the point to
H(+l), to K( 1), to H again, and so on indefinitely.
Clearly the two algebraic operations which, by definition,
must produce, when applied in this way, the sequence +1,
1, +1, 1, ..., represent a repeated multiplication,
either by +V 1 or by V 1.
Thus for exactly the same reason that we identify 1
with a unit step taken along a line in a reverse direction to
the unit represented by +1, we may identify +V 1 with
the revolution of a line through a right angle in one sense,
TOWARDS DE MOIVRE. IMAGINARIES
391
and V 1 with an equal revolution in the opposite sense.
This is the accepted interpretation.
OS is regarded as the / (or V 1) direction, and OT as
the i (or V 1) direction.
Complex Numbers
Revise the early work on the significance of coordinates.
Given a fixed line OA, and a fixed origin as at O, there
are two convenient ways of fixing the position of a point P.
1. Rectangular coordinates: Op 5, Pp = 2.
2. Polar coordinates: Z.AOP = 22, OP 54.
Evidently we may regard the rectangular coordinates as
specifying not merely measurements which define the position
of P, but also move n
ments by which P could
be reached from O. The
two movements would
be, one of +5 along
OA and one of +2 at
right angles to OA. .
The polar coordinates
specify much the same thing, though in a different way. If
to begin with we are at O and facing A, then the polar co
ordinates may be taken as instructions, first to turn through
an angle of 22, and then advance along OP a distance of 54.
If along a straight line a point takes two successive move
ments OA, AB, the length and
direction of OB is the algebraic
sum of the two movements.
If OA and OB are straight
lines or vectors which represent
two movements not in the same
straight line, the directed line OB
which closes the triangle OAB
may again be called the " sum " of OA and OB, since it
B
Fig. 236
392 CRAFTSMANSHIP IN MATHEMATICS
represents the single movement equivalent to the combin
ation of the two movements. Thus in fig. 225, OP may be
called the sum (more fully, the vector sum) of the movements
+5 along OA and +2 at right angles to OA. But if the
movement OP be represented by the symbol R, we cannot
in this case write R = (+5) + (+2), for this would repre
sent a movement of +7 from O along the line OA. But we
p may still represent R as
a sum, provided we do
/ \ something to indicate
that the component move
ments are at right angles.
Ui , 5 [p A F r this purpose the
Fig. 227 l etter * is prefixed to
that directed number
which represents the component at right angles to the initial
line. This is in accordance with our interpretation of A/ 1.
Thus the movement of OP would be represented by the
notation (+5) + i(+2). (Fig. 227.)
Of course, if P is confined to the line OA, a single directed
number will suffice to define its position after a series of
movements. But if P is forced to move about over the whole
plane of the paper, its position may be fixed just as definitely
by such an expression as, say, (13) + *'(+21).
Thus we may regard an expression of the form a + ib
as a complex number which serves to fix the position of a
point in a plane, just as the simple number a or b fixes its
position in a straight line.
But bear in mind that the term " complex number " is
only a convenient label, suggested by analogy; a + ib is
not really one number but a combination of two numbers,
together with a symbol i which stands for no number at
all. The symbol i is merely a direction indicator to
show that the movement or measurement represented by the
second number of the complex number is at right angles
to that represented by the first.
Let a, b be the rectangular coordinates, and r, a the
TOWARDS DE MOIVRE. IMAGINARIES
393
a
Fig. 228
polar coordinates of a point P. Then, since a = r cosa
and b = r sina,
a + ib = r cosa + i(r sina).
Again, let P' be the point on
OP at unit distance from O.
Then the movement OP' may
be represented by the complex
number, cosa + / sina. But
since r steps, each of length OP', would carry a point from
O to P, we may write:
a + ib = (cosa + /sina) X r,
or, more conveniently,
a + ib = r(cosa + i sina).
It follows that we may write:
(r cosa) f i(r sina) = y(cosa J i sina).
The conclusion is important, for it shows that we may, at
least in this connexion, proceed just as if i stood for a number,
Otherwise we could not legitimately assume that the twc
expressions are equivalent.
Note that the nondirected number r is called the modulus
of the complex number a + ib, and the angle a its amplitude
The operation which carries OA from its original position
to OB, then to OC, then to OP, in equal jumps, may be
looked upon as the repetition of a con _ p
slant factor, viz. a factor of the form
cosa + V^OL sina, i.e. cosa f / sina,
where a is the constant angle between
the rays from O. Since two rays
divide the /.AOP into three equal
parts, we may infer that m 1 rays
would divide it into m equal parts.
Fig. 229
394 CRAFTSMANSHIP IN MATHEMATICS
Hence, if ZAOP = 0,0 = ma. Since OA = r, the line OP
may be represented by the expression:
r(cosO J i sinO)
(or, by a + $, where a = r cos# and b = r sin#).
Again, since the factor cosa + i sina represents the turning
of the line from its original position OA through the angle
a, the factor
(cosa j tsina) x (cos(3 + tsinp)
must, presumably, represent a turning through the angle
(a + j8), and therefore be equivalent to the factor
cos (a + P) + i sin (a + P).
Obviously, then, the identity
coswa j i sin;wa (cosa + tsina) w
is foreshadowed. The usual sequel is obvious and simple.
Practice in the addition and subtraction of complex
numbers is desirable; it is quite easy. Devise examples to
enforce the notion that i is just a direction indicator, pro
viding us with a simple means of fixing a point P anywhere
in a plane containing an initial line; that it serves to show
that the second element of a complex number is at right
angles to the first. Practice in multiplication and division
should follow; this is also quite easy, once the boys see that
cosa + i sina is merely a " direction coefficient ", i.e. a
complex number which, when it multiplies another number,
produces a result which corresponds to the turning of a line
through the angle a. De Moivre easily follows.
The term " imaginary number " is not a happy one;
V 1 is just a symbol which can be treated in certain cases
as if it were a number. In the complex number a + 16,
a is often called the real, and ib the imaginary part.
The fruitful suggestion was made by Gauss that instead
of calling +1, 1, and V 1, positive, negative, and
imaginary units, we should call them direct, inverse, and
TOWARDS DE MOIVRE. IMAGINARIES 395
lateral units. To Gauss the radical difference between a
complex number and a rational number was that while the
latter denotes the position of points along a line, the former
denotes the position of points in a plane.
a f AV 1 must be regarded as the typical number of
algebra, " real " numbers being merely special cases in
which b = 0. If we are confined to real values of the variables
in y = f(x), we must admit that in the case of most functions
there are either values of x to which no values of y correspond,
or values of y which are not produced by any value of x.
But if the variables are complex numbers, these exceptions
never occur. To a value of x of the form a + b\/ 1, there
corresponds, in the case of every possible function, a value
of y of the form A + BV 1, #> b, A, B being themselves
real numbers.
The principle is so important that it must be understood
thoroughly by all pupils. Emphasize strongly the fact that
real numbers correspond to points in a straight line, complex
numbers to points in a plane. If we represent the values
of x by points in one line, and those of y by points in another,
we cannot say that any function y f(x) establishes a one
toone correspondence between all the points on the two
lines; in most cases, whole stretches of points will remain
outside the correspondence. But if we take two planes,
and represent the values of x by the points of one of them,
and the values of y by points of the other, we then obtain,
in every function, a onetoone correspondence between all
the points in the two planes. This is the key to the secret
of quadratic equations with " imaginary " roots.
Quadratic Equations and (socalled) Imaginary
Roots
Complex numbers can be used to explain certain diffi
culties met with in the study of quadratic equations. Consider
the example x 2 Gx + 34 = 0; the roots of which are
396
CRAFTSMANSHIP IN MATHEMATICS
sometimes said to be 3 V 25. But x 2 &x + 34,
i.e. (x 3) 2 + 5 2 , cannot be factorized; hence (we usually
argue) there is no value of x for which y (in y = x 2 6#
+ 34) is zero; in other words, the equation has no real
roots. Another way of stating this is that the parabola
y = x 2 6x + 34 has no points below y = 25 and there
fore does not cross the axis of x. Here is a graph of the
function:
(2
\
70
/
,6
,\
/
\
\
_>
i40
/
/
(
),3<
i\
30
\
J
29
)
X
^^ \ u ~*>
y
?0
(3,2
^
10


53 O ^ +5 +IO
i
Fig. 230
#
34
X 
3
O
+ 3
+ 5
HO
re* 
9
O
9
25
IOO
GX
id
O
18
30
60
34 
34
34
34
34*
34
Y 
61
34
25
29
74
X
But if i be treated as a number whose square is 1, we may
write,
(x  3) a + 5 2 = (x  3) 2  f* . 5 2
= (?  3 f 5/) (*  3  50.
Apparently, then, y = if jc = +3 5i.
It is usual to say that these values are " imaginary roots "
of the equation, or that they describe imaginary points where
the parabola may be supposed to cross the axis of x.
But from what we have already said about the nature of
i, there is clearly an alternative way of regarding this, a way
much more rational. The values +3 5i describe points
not on the axis of x, but elsewhere in some plane containing
TOWARDS DE MOIVRE. IMAGINARIES 397
that line. It is obvious that it cannot be the plane of the
paper, and we must therefore look for points in the plane
which is at right angles to the plane of the paper.
The necessary figure (231) consists of two parabolas, each
y = x 2 6# + 34, head to head, with a common axis but
in two planes at right angles to each other. A suitable sketch
is a little difficult to make, but it may be done in this way.
Let ABCD, EFGH be a rectangular block with square ends.
Bisect the block by the midperpendicular planes JKLM,
NPQR, STUV. The first and second intersect in the line
ab, of which V is the midpoint. In the horizontal plane
fliLM, draw the parabola y = x* fix + 34, with vertex
at V. In the vertical plane o/TS, draw the same parabola,
also with its vertex at V. The line mVn is the common axis
of both parabolas. The heavy lines in the plane JKLM
(xQx and Oy) are the coordinate axes of the primary parabola
in the horizontal plane. The axis of the parabola intersects
the x axis in z. As in the previous figure, Oz = 3, zV 25.
If, instead of y = (x 3) 2 + 5 2 the parabola was
y = (# 3) 2 , the parabola would touch the axis of x at z
(= + 3), but when the parabola moves into the position
y = ( x 3)2 _j_ 52^ i ts vertex is at V, (5) 2 units from the
x axis. Hence, the points given by the complex values of
x answering to y are at a distance 5 above and below the
plane of the primary parabola, and on a similar parabola to
the first, viz. the parabola in the vertical plane. Evidently
the points are on a line through z, m' , and n' y each 5 units
from z.
Thus, when we take into account complex values of
x, the complete graph corresponding to real values of the
function y (x 3) 2 + 5 2 is not one parabola but two,
lying in two planes. The parabola in the perpendicular plane
contains all points answering to complex values of x which
satisfy the given relation. Figs. 230 and 231 should be
compared.
Note that the line y = 25 lies in the plane NPQR, which
is tangential to both parabolas.
398 CRAFTSMANSHIP IN MATHEMATICS
TOWARDS DE MOIVRE. IMAGINARIES 399
Note also the difference between these two equations:
* 2  6*  16 = 0.
x 2 6x + 9 = 25,
3 + 5.
* 2  6* + 34 = 0.
/. x 2  6x + 9 = 25,
/. (96  3) 2 = + V 25 = + 16,
x = 3 + *5.
(V25)
Fig. 233
To obtain the points on the
curve we proceed from the origin
to z, f3 units away, in the x axis,
and then, also in the x axis, we
proceed from z, +6 and 5 units,
and so reach the points +8 and
2. The vertex is 25 units below
the x axis (see fig. above).
The journey is a journey in
one line, the x axis. The two
5's are measured from z.
To obtain the points on the
curve, we proceed from the origin
to s, 3 units away, in the x axis,
and then we proceed +i5 and
i5 units from ar, i.e. +5 and 5
units in a plane perpendicular to
the plane of the parabola, where
we reach points on a similar para
bola in this new plane.
The vertex of the primary
parabola is 25 units above the x
axis (see fig. above and fig. 231).
The journey is a journey in
two lines perpendicular to each
other. The two 5's are measured
from z as before, but in a perpen
dicular plane.
400 CRAFTSMANSHIP IN MATHEMATICS
Unless provided with a wire model (fig. 231), or with a
really good perspective sketch, boys are apt to be puzzled
by this problem. A model is much to be preferred; then
the effect of increasing and decreasing the distance of the
primary parabola from the x axis is easily observed.
Warn the boys not to be led away by the remarkable
and perfectly logical consistency of the hypothesis concerning
V 1. It is only an hypothesis after all. Still, it is not
advisable for learners to talk about " imaginary " roots of
equations but rather to explain such roots in the light of
the hypothesis in question.
We have touched upon vector algebra. The subject
receives considerable attention in Technical Schools but
very little in Secondary. This is a pity, for it is a cunningly
wrought instrument and is as useful as it is illuminating.
Quite the best introduction to it is Part I (Kinematic) of
Clifford's Elements of Dynamic. The first two parts of the
book, Steps, and Rotation, should be read by all teachers of
mathematics, and the third part, Strains, by all teachers of
mechanics. Maxwell's Matter and Motion is a little book
dealing admirably with the same subject. Henrici and
Turner's Vectors and Rotors is also useful.
TOWARDS THE CALCULUS 401
CHAPTER XXIX
Towards the Calculus
Coordinate Geometry
Teachers differ in opinion whether the calculus should
be preceded by a course of coordinate geometry. Certainly
anything like a complete course of coordinate geometry is
not a necessary preliminary. On the other hand, some little
knowledge of its fundamentals is advisable, and this is easily
developed from the previous knowledge of graphs. The
notion of the differential coefficient is nearly always made
to emerge from considerations of the tangent to the parabola,
but, more frequently than not, the common properties of
the parabola have not previously been taught. This partly
explains the haze which often enshrouds the notions under
lying the new subject.
A minimum of preliminary work in coordinate geometry
may be outlined.
The boys already know that y mx + c represents a line
making an angle whose tangent is m with the axis of x\ that, in
short, m represents the slope of the line; and that, in whatever
other form the equation may be written, it may be recast
into the y mx form, and its slope be determined at once.
X *V
For instance, the intercept form ~ + ~ = 1 mav be written
3
y = ^/%x f 3, and the slope is seen to be \/3.
The boys must be able to determine the equation of a
line satisfying necessary conditions. They already know that
if they are told a straight line must pass through a given
point, this condition alone is not enough to determine the
line, since any number of lines may pass through the point.
But if they are given some second condition, e.g. the direction
of the line, or the position of a second point through which
(291 27
402
CRAFTSMANSHIP IN MATHEMATICS
it passes, then the two data completely fix the line. This
fits in with the fact that the equation of the line must contain
two constants. Make the boys thoroughly familiar with the
ordinary rules for finding the equation of a line satisfying
two given conditions.
Types of suitable exercises for blackboard oral work:
1. Find the equation of a straight line cutting off an
intercept of 2 units' length on the axis of y and passing through
the point (3, 5).
2. Find the equation of a straight line drawn through
the point (3, 5), making an angle of 60 with the axis of x.
3. Find the equation of a straight line passing through
the points (2, 3), (4, 1).
The last exercise is a type with which the boys should be
thoroughly familiar. It will be required often in future
work. The equation should therefore be familiar in its
general form, and should be illustrated geometrically.
Find the equation of a line passing through the two points
A(#i, jVi) and B(# 2 , y 2 ). We may proceed in this way:
WT
Suppose y = mx
m and c are unknown.
c represents the equation, where
The particular point (# t , y x ) is on the line, .'. y l mx l + c. (i)
The particular point (x 2 , y 2 ) is on the line, .'. y% = mx 2 f c. (ii)
The point (#, y), any point, is on the line, .'. y = mx + c. (iii)
TOWARDS THE CALCULUS 403
Subtracting (i) from (iii),
y y 1 = m(x xj.
Subtracting (i) from (ii),
^2 y\ = "K*2 *i)
.'. by division
or (y  yi ) 
which is the required equation, ^  ~ representing m
(the slope) in y = mjc. ^ 2 ~ ^
Beginners rarely see this clearly, unless the algebra is
clearly illustrated geometrically. From the last figure,
take out the two similar triangles, and show the lengths
of their perpendiculars and bases in terms of coordinates.
The slope of the line AB is given from the first triangle;
= 9* iJ, A and B being the two specified points on
AK # 2 ~" #1
the line.
The slope of the line AB is also given from the second
triangle,  = y ~~ ?\ A being a specified point, and P
AM x #!
being any point on the line.
404 CRAFTSMANSHIP IN MATHEMATICS
But the triangles are similar, and the slopes are therefore
identical.
. y ~~ yi = yLny\
X X l X 2 Xj*
or (y y x ) = ^"H^ 1 (x x^, as before.
# 2 Xi
Or again: take the slope found in the last case, say ,
and substitute for m in y ~ mx + c. We have * 2 ~~ x *
a)
and since the line passes through (x ly j/ x ), [or (# 2 , y 2 ) might
be chosen if preferred],
By subtraction, we have from (i) and (ii)
~ x \)t as before.
Every step must be substantiated geometrically. It is
fatal to allow the boys to look upon the problem as mere
algebra. The boys should be able to write down instantly the
equation of a line passing through two points and under
stand its full significance. But in evaluations of this kind,
do not be satisfied with just the typical textbook formal
solutions. Vary the work.
The Parabola and its Properties
Throughout the teaching of coordinate geometry, let all
principles be established first by pure geometry. Let the
picture come first, and teach its new lessons. Then let
symbolism follow. Geometry treated as pure algebra tends
to lose all semblance of its essential space relations. At
TOWARDS THE CALCULUS
105
least the parabola, if not the ellipse, will already have received
some attention. It will have been touched upon in connexion
with graphs and quadratic equations, and the boys may have
learnt something about the paths of falling bodies; they
may also know that in certain circumstances the chains of a
suspension bridge, and vertical sections of the surface of a
rotating liquid, form parabolas.
It is useful to give the boys a mechanical means of readily
drawing a parabola. It saves much time, and encourages
them to use good figures. Here is one way.
An ordinary Tsquare slides along AB, the lefthand edge
of a drawing board, in the usual way, the edge AB answering
C
Fig. 235
as a directrix. A string equal in length to KG is fastened
at G, and at a fixed point S in a line XX perpendicular to
AB. A pencil point P keeps the string stretched and remains
in contact with the edge KG of the Tsquare. As the Tsquare
slides up and down the edge of the board, the pencil traces
out a parabola with focus S.
A parabola is the locus of a point whose distance from a
406
CRAFTSMANSHIP IN MATHEMATICS
fixed point is equal to its distance from a fixed straight
line. Help the boys to see that from this definition certain
properties follow at once:
(1) PM (diam.) = PS.
(2) ES (semilatus rectum) = EM' = SX.
(3) AS = AX.
(4) SX = 2SA.
(5) ES  2SA.
(6) EF (latus rectum) = 4SA.
The main property to be mastered is the slope of the
tangent, and to this end the following summary of pre
liminary work is suggested. All principles should be established
first geometrically, then
 analytically, and the
boys must be made to
see that the results are
identical.
1 . The principal
ordinate of any point P
on a parabola is a mean
proportional to its ab
scissa and the latus rec
tum.
i.e. PN 2  4AS.AN.
Analytically: call the
point P, (x 9 y)\ SA =
Fig. 236 AX = *(8ay). Theny 2
 4ax. (Fig. 236.)
If the directrix MX is the y axis, the equation becomes
y 2 = &a(x a).
2. If a chord PQ intersects the directrix in R, SR bisects
the external angle QSP' of the triangle PSQ.
TOWARDS THE CALCULUS
407
Drop JLs PM
and QV on direo
trix. As PMR and
QVR are similar.
PR
QR
PM
QV
PS
QS'
Fig. 237
3. If the tangent
at P meets the direc
trix in R, the angle
PSR is a right angle.
Deduce this from
the preceding pro
position. When Q
coalesces with P,
each of the marked
equal angles becomes a right angle. (See next figure.)
4. The tangents at the extremities of a focal chord intersect
D
PP jbcal chord
RP, RP  tangents
Z.PRP*
Fig. 238
at right angles on the directrix, i.e. the tangents at P and P',
the extremities of the focal chord PSP ; , make a right angle
408
CRAFTSMANSHIP IN MATHEMATICS
at R, where they meet on the directrix. Observe that RS
meets the focal chord at right angles (cf. No. 3).
5. The subtangent is equal to twice the abscissa, i.e. TA= AN
or, TN = 2AN.
Fig. 239
6. The foot of the focal perpendicular of any tangent lies
on the tangent at the vertex, i.e. Y, the foot of the L SY to
the tangent PT, lies on the tangent at A.
Fig. 340
TOWARDS THE CALCULUS 409
rr>i 7 r . .2 latus rectum ~
7. 1 he slope of any tangent =     . ror m
^ J J 6 ordinate
the last figure the triangles YAS and TNP are similar. Hence
slope of tangent PT
^ PN = SA = 2AS ^ \ latus rectum
"" NT ~ AY ~~ PN ordinate '
i.e. the slope of the tangent to the axis of the parabola. If
the figure is turned round, so that the slope of PT is to the
ATT i 11 ordinate
tangent AY at the vertex, then slope =   .
$ latus rectum
How may this slope be expressed in rectangular co
ordinates? The equation of a secant cutting the curve in
p *> and Q is
If a figure be drawn accurately (this is not easy), actual
measurement will show that the slope of the secant is the
ratio
\ latus rectum^
mean of ordinates
To obtain the equation of the tangent at (x lt y^ we take
Q indefinitely close to P, so that ultimately j> 2 = y v The
equation to the secant then becomes:
a, ,
or y = (x + *!
which is the equation to the tangent, and the slope of the
* *u 20 . 4 latus rectum u 
tangent is thus , i.e. =    , as before.
y l ordinate
410 CRAFTSMANSHIP IN MATHEMATICS
The Tangent to the Parabola
If future work is to be understood, the tangent to the
parabola, and its various implications, must receive close
attention. The necessary further elucidation may thus be
summarized.
1. To find the condition that the straight line y = mx f c
may touch the parabola y 2 * kax.
Since y = mx f c,
.'. y 2  (mx + c)*;
and since y 2 lax,
:. (mx + f) 2 = lax.
By solving this equation we obtain the abscissae of the two
points in which the straight line cuts the curve. The line
will touch the curve if the two points coincide, and the con
dition for this is that the roots must be equal,
i.e. in m 2 * 2 + 2x(mc  2a) + c 2 = 0,
l(mc  2a) 2 = Im 2 c 2 9
a
i.e. a = me, or c .
m
Hence the line y = mx + c touches the curve y 2 kax if
c = _ (where m is the slope which the tangent makes with
m
the axis).
2. To find the point where the tangent y = mx + touches
the parabola y 4a#.
As before,
(mx + c) 2 = lax,
/. [mx j ) == 40#,
> m'
(
or \ mx 1 =
and since y 2 = lax, y = .
TOWARDS THE CALCULUS 411
a 2a\
Hence the point required is ( , ).
\m 2 m/
3. Compare the two forms of the equation of the tangent,
viz. yy = 2a(x f #i)> and y = mx + .
m
The first may be written,
2a
y = x
Hence we may write the two forms in parallel thus:
y = mx + 
nt
TU * *u r . a ,
They represent the same line; . . m, and
Ji i
That these two last equations are consistent may easily be
shown by evolving the second from the first, the connecting
link being y^ =
<0 t_
Evidently, then, the equation y = mx + 1S the tangent
o ^^
at the point (x l9 y^, i.e. ( , ).
\m m /
4. Verify geometrically that the tangent y = w# +
9 Wl
touches the parabola y 2 = &ax at the point ( , j. (This
verification is of great importance.) m
(The abler boys ought to do this without any further
help).
AY = tangt. at vertex = y axis.
AX = x axis.
S = focus.
TA = AN (subtangent = 2 abscissa)
SZ meets the tangent PT at rt. Z.s at Z, since AZ is the tangent at
vertex.
412
CRAFTSMANSHIP IN MATHEMATICS
The rt. angled As TAZ, ZAS, TZS, TNP are all similar.
Since TA = TN, /. ZA = PN.
AS = a = dist. of focus to vertex.
Fig. 241
AZ = = intercept on y axis.
m
PN
 = m slope of tangent.
TA _ AZ _. _ AZ 2
_= _,or I*,
/. AN
AZ 2
AS
a
= abscissa of P,
and
2a
PN = 2ZA = = ordinate of P.
Q.E.D.
Observe, again, that the slope of the tangent
__ \ latus rectum __ 2a
ordinate
 = m.
2a
m
This pictorial parallelism between the geometry and
algebra is essential whenever it is possible. Let the boys
see that coordinate geometry is geometry and not mere
algebra. But of course the geometrical figure is also a
graph, to be interpreted algebraically.
We have taken the subject of coordinate geometry
TOWARDS THE CALCULUS 413
thus far, less for its own sake than as an introduction to the
next chapter. Coordinate geometry is an easy subject to
teach, and boys like it, provided the geometry itself is made
clear. As a subject of mere algebraic manipulation, un
associated with pure geometry, its value is slight, and time
should not be spent over it.
Methods of Approximation
The calculus is such a valuable mathematical weapon,
and the fundamental ideas underlying it are so simple, that
the subject should find a place in every Secondary school.
It might be begun in the Fifth Form, if not in the Fourth,
though naturally the first presentation must be of a simple
character. This simple presentation is easily possible. The
more technical side of the subject, as elaborated in the
standard textbooks, is wholly unnecessary in schools.
It was, I believe, Professor Nunn who pointed out that
the history of the subject suggests the best route for teachers
to follow. Although Newton and Leibniz are rightly given
the credit of being the creators of the calculus as a finished
weapon, the preliminary work of certain of their predecessors,
especially Wallis, from which the main idea of the calculus
was derived, must always be borne in mind. Wallis's work
is merely a special kind of algebra and may readily be under
stood by a welltaught Fourth Form.
If we are thus to begin with approximation work, there
is much to be said, as pointed out by Professor Nunn, for
beginning with integration rather than with differentiation.
The necessary arguments are so simple and the results
so valuable that the rather radical departure from normal
sequence is justified. For all practical purposes, Wallis
was the actual inventor of the integral calculus, and Wallis 's
own work and methods serve to give young pupils a clear
insight into the new ideas.
This early work, in differentiation as well as in integra
tion, should be taught as a calculus of approximations. The
414 CRAFTSMANSHIP IN MATHEMATICS
pupils should learn that such investigations give results
which may be regarded as true to any degree of approxima
tion, though not absolutely true. When later the calculus
itself is formally taken up, and the pupils are able to grasp
the modern theory of " limits ", they should be able to see
that the new arguments, if properly stated, do as a matter
of fact give results which are unequivocally exact. They
must not be allowed to assume, at that later stage, that the
arguments of the calculus prove merely approximately true
results, and yet that these may be treated as if they were
exact truths. This illegitimate jump from possible truth
to certain truth is often made, it is true; but the deduction
commonly involves the fallacious use of such terms as
" infinitely small ", " infinitely great ", and the like.
" Methods of approximation are inferior methods and
do not yield exact results." Granted. But these methods
are best for beginners, if only because they form a good
introduction to the exact methods of the calculus, and they
are, after all, based upon a kind of reasoning which is rigor
ous enough for practical purposes. But the important thing
is to make the pupils feel that they must never be finally
satisfied until they have mastered a method which yields
results that admit of no doubt at all.
The beginner has already learnt, or should have learnt,
from his graph work the main idea of the real business to
be taken in hand, and that is the nature of a function: that
the value of one variable can be calculated from the value
of another by the uniform application of a definite rule
expressed algebraically.
We append a few suggestions for work in suitable approxi
mations.
1. Rough approximations.
(i) Revise certain exercises in mensuration, e.g. find the
area of a circle and of some irregular figures by the squared
paper method.
(ii) Surveyor's FieldBook exercises; measure up some
TOWARDS THE CALCULUS 415
irregular field, or other area, but insist that all such results
are only rough approximations.
2. Closer approximations, and the methods involved.
(i) Revise the work on expansion (in physics). For instance,
the coefficient of linear expansion of iron is '00001. Justify
the rule of accepting 00002 instead of (00001) 2 for area
expansion, and 00003 instead of (00001) 3 for cubical expan
sion. Show the utter insignificance of the rejected decimal
places. Refer to the geometrical illustrations of (a + b) z
and (a + ft) 3 .
(ii) Estimate the area of a triangle as the sum of a number
of parallelograms. The more numerous the parallels and the
z w .
Fig. 243
more numerous the parallelograms, the more negligible do
the projecting little triangles become. Observe that if the
number of parallelograms is doubled, each shaded triangle
is reduced to onefourth; and so on.
(iii) Estimate the volume of a pyramid as the sum of the
volumes of a number of flat prisms, gradually diminished
in thickness.
(iv) Estimate the volume of a sphere regarded as the sum
of a number of pyramids formed by joining the centre of the
sphere to the angular points of a polyhedron, the number of
whose faces is increased indefinitely. The pyramids formed
have as their bases the faces of the polyhedron; the volume
of each pyramid (face X height) /3, hence the volume of
the sum of the pyramids = (sum of faces) X height/3. If
the number of faces be increased, the sum of the faces becomes
more nearly equal to the area of the spherical surface, and
then the height of the pyramid is more nearly equal to the
416 CRAFTSMANSHIP IN MATHEMATICS
radius r of the sphere. But the sum of the faces can never
be quite equal to the surface of the sphere, though we can
so increase the number of faces of the polyhedron that the
approximation may be closer than any degree we like to
name. The spherical surface is necessarily greater than the
sum of the flat faces of the polyhedron and can never be
reached: it is an unreachable limit. If the sum of the faces
could become equal to the surface of the sphere the sum
would be 4:irr 2 and then the height of each pyramid would
be equal to r. Hence the volume of the sphere would be
r 4
4?rr 2 X  = 77T 3 . Now this result agrees with the result
3 o
arrived at by other methods, and it is correct. Still, to obtain
the result, we had to jump from flatfaced pyramids (though
these may have been made inconceivably small) to corre
sponding bits of spherical surface which were not flat.
We have still to discover whether such a method is allow
able.
(v) The value of TT. The pupils may be allowed to assume
from a figure they will readily guess that if 2 regular
polygons with the same number of sides be, respectively,
inscribed within and circumscribed without a circle, the
length of the circumference of the circle will be less than
the perimeter of the circumscribed polygon but greater than
that of the inscribed polygon. Show the pupils that the
determination of TT is thus merely a question of arithmetic,
though of very laborious arithmetic, inasmuch as we have to
determine the perimeters of polygons of a very large number
of sides; the greater the number of sides, the greater the
degree of approximation of the value of TT. Give a short
history of the evaluation of TT, from the time of Archimedes
onwards. Point out that the irrationality of TT has now been
definitely demonstrated, so that it is useless for computers
to waste any more time over it. Make the pupils see that
the method of evaluating TT is only a method of approximation,
and that in this case no better method is ever to be hoped
for; that we can obtain values more and more approximating
TOWARDS THE CALCULUS
417
to the ratio of the circumference to the diameter, but there
can be no final value, as the decimal can never terminate.
Area under a Parabola
We cut off a parabola by a line P'P perpendicular to its
axis OA, and enclose it in the rectangle P'M'MP. We will
calculate the shaded area OPM, i.e. the area " outside " or
" under " the half parabola AOP. Let the parabola bey=kx 2
We may divide OM into any number of equal parts, and
on these parts construct a number of rectangles of equal
breadth, set side by side as shown in the figure. The added
areas of the rectangles are evidently in excess of the area
OPA, but by increasing the number of rectangles indefinitely,
the excess is indefinitely diminished.
We will begin by dividing OM into a small number of
parts, and then increase the number gradually. As the first
division OQ is gradually to be diminished, we will consider
the rectangle on it to be of zero area. Hence OR = (2 1)
= 1 unit; .'. RR' = I 2 units, OS = (3 1) = 2 units;
(K291) 28
418 CRAFTSMANSHIP IN MATHEMATICS
A PA
L
~\ I
TOTAL ARC* AM .
DIV DEO INTO 43 2
as.
TOTAL AREA AM
DIVIDED INTO 3*2*
12
2*
 4 '
. . _, _ s
.av.
R
z
I 2
L Z
___ TOT
M
2 O*i25 0123*
P A
TOTAL AREA A M
TOTAL AREA A M
TOTAL AREA A M
DIVIDED INTO 6*5
DIVIDED INTO 7 "6
DIVIDED INTO 8 X 7
 150
252
1
39E
j
2
6
7
1
2
2
5
G'
2
2
I 3
FT
4
,
5
II
i.t
1 *
4
i 1 2 1
[?[
P] 5
D9RSTUM 09RSTUVM O9RST
U V W M
4567
(IV)
(v)
Fig. 244
.'. SS'  2 2 units; OT = (4  1) = 3 units, /. TT' =
units; &c.
We may tabulate the results thus:
Linear
Units in
Square Units of Area in
Ratio of
(*) to (a).
OM.
PM.
(a) Rect.
AM.
(6) Sum of contained Rectangles.
Lowest
Terms.
Re
written.
3
2 2
3 x 2 2
1 2 + 2 2
A
A
4
3 2
4 X 3 2
I 2 + 2 2 + 3 2
A
A
5
42
5 X 4 2
I 2 } 2 2 f 3 2 + 4 2
a
8
9
24
'6
5 a
6 X 6 2
p + 2 2 + 32 + 42 + 52
M
a*
7
6 2
7 X 6 2
I 2 + 2 2 + 3 2 f 4 2 f 5 2 + 6 2
M
11
8
7 2
8 X 7 2
I 2 + 2 2 + 3 2 f 4 a + 5 2 + 6 2 + 7 2
A
M
TOWARDS THE CALCULUS 419
The rewritten ratios show the numerators in A .P., and the
denominators as multiples of 6. Obviously if m be the
number of units in QM, the ratio may be written  or
1 Gm
Thus the area of the added rectangles is equal to the
area of the rectangle AM + a fraction depending on the
value of m. It is easy to prove that the law holds good for
all values of m.
The ratio i _ enables us to write down as many
6m J
terms as we please. For instance if m 1000, the ratio
= elH) o or 4 + 6 oW Hence if we built up a figure with
1000 rectangles, the total area of the rectangles would be equal
to of AM + a small area equal to Q^Q of AM.
Evidently by taking m large enough, the fraction
6m
becomes so small as to be insignificant, and thus the com
bined area of the rectangles can be made to differ as little
as we please from  the area of AM. And as the rectangles
are made narrower and narrower, the area they cover will
eventually become indistinguishable from the area under
the curve OP (fig. 243); e.g. if m = 1000 the sum of the top
lefthand corners of the rectangles projecting outside the curve
is only QQOQ ^ AM. Finally the tops of the rectangles will
be indistinguishable from the curve itself. We conclude,
therefore, that this area under the curve is, at least to a very
great degree of accuracy,  of the rectangle AM. Since
OM = x, and PM kx* (fig, 243), we express the conclusion
by the formula
A =
It follows that the area AOP is f the area AM. Hence the
whole area of the parabola up to PT is f OA X P'P.
A point for emphasis: " Having proved that the area
under the curve is, apparently to an unlimited degree of
closeness, \ of the rectangle AM, we are almost forced to
420
CRAFTSMANSHIP IN MATHEMATICS
believe that the former is exactly \ of the latter." Still, the
fact remains that what we have proved is only an approxima
tion. Do not disguise the theoretical imperfection of the
conclusion. Do not slur over the fact that we have merely
an approximation formula, though it is quite proper to em
phasize the other fact that no limit can be set to the close
ness of the approximation which it represents.
The particular approximation result arrived at is easily
extended. Let an ordinate start from the origin and move
to the right. If it has a constant height, y = k, it will, in
moving through a distance x, trace out an area, A = kx.
If its height is at first zero, but increases in accordance with
one of the laws y = kx, y = kx*, y  kx 3 , the area traced
out will be given by the corresponding law, A \kx*>
A = J&# 3 , A = \kofi. These results we might establish by
proceeding exactly as before. Calling the function which
gives the height of the ordinate, the ordinate function, and
the function which gives the area traced as the area function^
the results may be summarized simply
Ordinate functions
Corresponding area functions
kx
kx 1
1
kx 2
s*
kx 3
4 iX
kx n ~ l
~kx n
n
This summary exhibits Wallis's Law.
Books to consult:
1. Teaching of Algebra, Nunn.
2. Cartesian Plane Geometry, Scott.
3. Modern Geometry, Durell.
THE CALCULUS: SOME FUNDAMENTALS 421
CHAPTER XXX
The Calculus : some Fundamentals
First Notions of Limits
To boys the two terms " infinity " and " limits " are
always bothersome, and it is doubtful if the first ought to
be used in Forms below the Sixth. A misapprehension as
to the significance of both terms is responsible for much
faulty work, much fallacious reasoning.
What is a point? " A point is that which marks position
but has no magnitude." But how can a thing with no magni
tude indicate position? If it has magnitude, is it of atomic
dimensions, say 10~ 24 cm.? Or is it 10~ 10 of this? Obviously
if it has magnitude at all, a certain definite number side by
side would make a centimetre. But this is entirely contrary
to the mathematician's idea of a point.
If a line is composed of points, the number of points
certainly cannot be finite; otherwise, if the number happened
to be odd, the line could not be bisected. Again, if the side
and the diagonal of a square each contained a finite number of
points, they would bear a definite numerical ratio to each
other, and this we know they do not. The existence of in
commensurables proves, in fact, that every finite line must,
if it consists of points, contain an infinite number. In other
words, if we were to take away the points one by one,
we should never take away all the points, however long we
continued the process. The number of points therefore
cannot be counted. This is the most characteristic property
of the infinite collection that it cannot be counted.
Consider two concentric circles. From any number of
points on the circumference of the outer circle, draw radii
to the common centre. Each radius cuts the circumference
of the inner circle, so that there is a onetoone correspondence
between all the points on the outer circle and all the points
422 CRAFTSMANSHIP IN MATHEMATICS
on the inner. Imagine the outer circle to be so large as to
extend to the stars, and the inner one to be so small as to
be only just visible to the naked eye. Further, imagine
an indefinitely large number of points packed closely round
the circumference of the big circle, and all the radii drawn;
the number of corresponding points on the inner circum
ference must be the same as the number on the outer.
Clearly, in any line however short, there are more points
than any assignable number. However large a number of
points we imagine in a line, no one of them can be said
to have a definite successor, for between any two points,
however close, there must always be others.
Again, consider the class of positive integers. They may
be put into onetoone correspondence with the class of
all even positive integers, by writing the classes as follows:
123456
2 I 4 I 6 I 8 I 10 I 12 I
To any integer a of the first class there corresponds an integer
20 of the second. Hence the number of all finite numbers
is not greater than the number of all even finite numbers.
Evidently we have a case of the whole being not greater
than its part.
Thus we have another characteristic of classes called in
finite: a class is said to be infinite if it contains apart which can
be put into a onetoone correspondence with the whole of itself.
It is possible to imagine any number of sequences whose
numbers have a onetoone correspondence with all the
integers, for instance all the multiples of 3, or of 7, or of 97.
The characteristics of all such sequences are: (1) there is
a definite first number; (2) there is no last number; (3) every
number has a definite successor. Hence they must all be
supposed to have the same infinite number of members.
It is important to notice that, given any infinite collection
of things, any finite number of things can be added or taken
away without increasing or decreasing the number in the
collection.
THE CALCULUS: SOME FUNDAMENTALS 423
It will be agreed that the nature of an infinite number is
beyond the conception of an ordinary boy, and the boy
should not be allowed to use the term. Even the ordinary
teacher may ponder over the paradox of Tristram Shandy.
A man undertakes to write a history of the world, and it
takes him a year to write up the events of a day. Obviously
if he lives but a finite number of years, the older he gets
the further away he will be from finishing his task. If, how
ever, he lives for ever, no part of the history will remain
unwritten. For the series of days and years has no last term;
the events of the nth day are written in the nth year. Since
any assigned day is the nth, any assigned day may be written
about, and therefore no part of the history will remain un
written.
Neither Tristram Shandy nor Zeno is meat for babes, but
there are certain elementary considerations of number se
quences with which boys should be familiar.
" Number " in the more general sense means simply
the ordinary integers and fractions of arithmetic. All numbers
in mathematics are based on the primitive series of integers.
A fraction is, strictly speaking, a pair of integers, associated
in accordance with a definite law. This law enables us to
substitute for each single integer a pair of integers which
are to be taken as equivalent to it. Thus  is equivalent to 5.
In this way we get an infinite number of numerical rationals
of the same form.
Between any two numbers of the sequence of integers,
there is an infinite number of rationals. For instance, between
8 and 13 there is an infinite series of rationals, or between 8 and
9. Obviously, then, the rationals between, say, 8 and 9 form
a sequence that is endless both ways. Between 8 and 9 we
have, for instance, 85; between 8 and 85 we have 825; between
825 and 85 we have 8375; and so on indefinitely. Con
secutive fractions, that is, fractions between which, for example,
a mean cannot be inserted, are inconceivable, just as are
consecutive points in space. The integers 8 and 9 are the
424 CRAFTSMANSHIP IN MATHEMATICS
first numbers met with beyond the sequence. We call these
numbers 8 and 9 the upper and lower limits of the sequence.
There is no last rational less than 9 and no first rational
greater than 8. It is erroneous to say that the terms of a
sequence ultimately coincide with the limit. The limit is
always outside the sequence of which it is the limit.
Consider the sequence 2 y, 2 J, 2 J, . . . 2 ,
n
as n increases endlessly. Here successively higher integral
values form a sequence of rationals which constantly rise in
value but have no last term. There is, however, a rational
number, 2, which comes next after all possible terms of the
sequence. That is to say, if any rational number be named
less than 2, there will always be some term less than 2 
n
between it and the number 2. This is what is meant by
calling 2 the limit of the sequence.
A and B are a given distance apart, say 2". We attempt
to reach B from A by taking a series of steps, the first step
being half the whole distance; the second, half the remaining
distance; the third, half the still remaining distance; and
so on. When would B be reached? Obviously the answer is
never. For any step taken is only half the distance still
remaining. Thus the successive distances, in inches, are,
1, , J, J, y, and so on. However many of these distances
are added together, the sum would never be equal to the
whole distance, 2". It is thus absurd to talk about summing
a series to infinity. The limit, 2, is not a member of the
series; it is unreachable and stands, a challenger, right out
side the series , as a limit always does.
The example is well worth pursuing.
The sum of the series 1 + 1 + I + . . . + A = 2  Jj.
The sum of the series l + + +...+ = 2.
THE CALCULUS: SOME FUNDAMENTALS 425
We can, of course, take a number of terms of the series that
will be great enough to make the sum fall short of 2 by less
than any fraction that can be named, say less than 1/100,000.
We have to calculate the value of n so that may be equal
to or less than 1/100,000. By trial we find = and
11 2 16 65536
2" = 131072* The lattef is leSS than 1 / 100 ' 000  Thus if
we take 17 terms of the series, the sum differs from 2 by
less than 1/100,000.
It is impossible to take enough terms to make the sum
equal to 2. There is always a gap l/2 n . However great n may
be, l/2 n is always something; it is never zero. We may say
that the sum of n terms becomes nearer and nearer 2 as n
becomes greater and greater, or that it tends towards 2 as n
becomes indefinitely great. Do not use the term infinity.
Consider another example, the decimal 11111....
11111... = ye + Too i" Tooo" ~r Too (To ~i~ ^ c '
The sum of the first 2 terms = = i .
100 9^
The sum of the first 3 terms =
The sum of the first 4 terms
1000 9 T  T *
1111 _ 1
10000 ~ 9'
By increasing the number of terms, the sum can be made to
differ by less and less from , and this difference can be made
smaller than any quantity that can be named. Hence ^ is
the limit to which the sum tends, though this limit can never
be actually reached.
So in cases like the area under a curve. Where we say that
PM is the limit of the ordinate pm, we mean that by taking
mM constantly smaller, pm may be brought constantly
nearer PM, and that it never occupies a position so near that
426 CRAFTSMANSHIP IN MATHEMATICS
it could not be still nearer. It always remains the opposite
side of the rectangle, and never actually coincides with PM,
for PM stands outside all pos
sible positions of pm. But PM
is the first ordinate that stands
outside the series, that is, there
is no other ordinate between
PM and the series of all possible
positions of pm.
Fig 24S The teacher should devise
other illustrations of the nature
of a limit. The notion is fundamental, and the pupils must
understand it.
Rate
The old geometers were concerned more with drawing
tangents to curves, and with finding the areas enclosed by
curves, than with rate of change in natural phenomena;
but the latter idea as well as the former one was certainly
in Newton's mind, and was embodied in the language of the
calculus, as we now call it, which he and Leibniz invented.
The two ideas, tangency and rate, are virtually just two facets
of the same idea, and in teaching the calculus the two should
be kept side by side.
Pupils will have learnt something already about tangency.
And if they have begun dynamics, as they ought to have done,
they will have some idea of the nature of " Rate ". Rate is
one of those rather subtle terms which are much better con
sistently used than formally defined. Even in the lower Forms,
boys should be given little sums in which the term is correctly
used: " at what rate was the car running?" and so forth.
But before the calculus is begun, the notion of rate must be
clarified. This means presenting the notion, in some way,
in the concrete. Practical work is essential, even if the experi
ments are only of a rough and ready character. Suitable
experiments are described in any modern book on dynamics.
Consider a train in motion. How can we determine its
THE CALCULUS: SOME FUNDAMENTALS 427
velocity at some instant, say at noon? We might take an
interval of 5 minutes which includes noon, and measure how
far the train has gone in that period. Suppose we find the
distance to be 5 miles; we may then conclude that the train was
running at 60 miles an hour. But 5 miles is a long distance,
and we cannot be sure that exactly at noon the train was
running at that speed. At noon it may have been running
70 miles an hour, or perhaps 50 miles, going downhill or
uphill at that time. It will be safer to work with a smaller
interval, say 1 minute, which includes noon (perhaps
half a minute before to half a minute after Big Ben begins
to strike), and to measure the distance traversed during
that period. But even greater accuracy may be required:
one minute is a rather long time. In practice, however, the
inevitable inaccuracy of our measurements makes it useless
to take too small a period, though in theory the smaller
the period the better, and we are tempted to say that for ideal
accuracy an " infinitely small " period is required. The older
mathematicians, Leibniz in particular, yielded to this tempta
tion, and so gave wrong explanations of the working of the
new mathematical instrument (the calculus) which they
invented.
Revise rapidly some of the easier
graph work and show how change of
rate is indicated by change of steep
ness in the slope.
The careful study of a falling body
will go far to make clear the notion of
rate. Refer to Galileo's experiments
on falling bodies. Generally speaking,
it will not be possible to repeat such
experiments, and so obtain firsthand
data; the necessary data must therefore
be provided otherwise. Let fig. 246# represent the path of
a body falling from a tower or down a well. The three
lines allow the three sets of values (distances, velocities, times)
to be shown in parallel, the distances and velocities being
DISTANCES VELOCITIES TIMES
(FT) (Frpersc.) fseca)
.64 64
256 1(28 J
Fig. 2460
428
CRAFTSMANSHIP IN MATHEMATICS
DISTANCES VELOCITIES TIME
(Fr) (FT parse^ (sec,)
shown at the end of 1, 2, 3, and 4 seconds, respectively. Use
the data to verify (perhaps in some degree to establish) the
formulae v = ft , v = M + /*> * = i(# + *0*> 5 = /* 2
Boys are often puzzled about the 32 (the acceleration
constant). In the first place, it is a power of 2, and they confuse
it with t 2 . (It is really best to use the nearer value 322,
even though the arithmetic is a little more difficult.) In the
second place the boys are apt to forget that this acceleration
number is merely the value attached to a particular interval
of time, viz. 1 second. They should be given a little practice
with smaller intervals, say J seconds.
The second figure, a modification of the
previous figure, is therefore useful. It
represents the happenings in the first
second, at quartersecond intervals.
Since the same amount of extra velocity
is added on per second, we have to take
onequarter of this for each quarter of
a second. Observe that although this
figure really represents the happenings
in the first second of the previous figure,
the two figures have an identical appear
ance so far as the linedivisions are con
cerned. The one second is divided up exactly as the four
seconds were divided up. It impresses boys greatly that this
sort of magnification or photographic enlargement might go
on " for ever ". If, for instance, we take the first quarter
second of the last figure (2466), and magnify the distance
line 16 times (as we did in the case of fig. 246a), we get still
another replica, this time with the quartersecond divided
up to show the happenings during each sixteenthsecond.
However short the distance, there is acceleration, and the
acceleration has a constant value. The acceleration is
" uniform ".
" Uniform acceleration is measured by the amount by
which the velocity increases in unit time." Many boys
have difficulty in understanding what " uniform " accelera
Fig. 2466
THE CALCULUS: SOME FUNDAMENTALS 429
Fig. 247
tion, such as acceleration due to gravity, really implies.
" If only you would accelerate by adding on velocity in
definite chunks at equal intervals, we could understand it."
Let the boy have his definite chunks, at first, and utilize
these for approaching the main idea. Go back to the graph.
Suppose a train to move for 1 minute at a uniform velocity
of 5 miles an hour; then to be suddenly accelerated to 10
miles an hour and to travel for 1 minute
at that velocity; then to be accelerated to 50
15 miles an hour for a third minute; to 20
miles an hour for a fourth minute; to 25 for
a fifth; and to 30 for a sixth. How far would
it have travelled altogether? A velocity
time graph shows this at once. The
number of units of area under the graph is
1 + 2 + 3 + 4 + 5 + 621, and this
gives us the number of miles travelled.
The dotted line passing through the top lefthand corners
of the rectangles can easily be proved to be straight, and
this evidently indicates some sort of uniformity in the motion.
But the whole of the area under this line is not enclosed by
the rectangles; there are 6 little triangles unaccounted for.
How are these triangles to be explained? By the fact that
really we have imagined an impossible thing, viz. that at
certain times the train's speed was instantaneously increased
5 miles an hour.
Now although in practice we know that even in the very
best trains acceleration is really brought about by sudden
jerks, these jerks are virtually imperceptible, and it is there
fore not impossible to imagine an acceleration free from such
sudden increases. It may be easily illustrated by running
water: the following ingenious illustration we owe to Professor
Nunn.
Attached to my bath is a tap so beautifully made that
by means of the graduated screwhead I can regulate the
amount of water running in up to 8 gallons a minute.
430
CRAFTSMANSHIP IN MATHEMATICS
I turn on the tap for one minute, the water running at
the rate of 1 gallon a minute; in that time 1 gallon has been
delivered. Then I turn the tap on further, to deliver water
at the rate of 2 gallons a minute, and allow it to run for one
minute; during this minute, 2 gallons have been delivered.
Thus I continue for 8 minutes, 8 gallons running in during
the eighth minute. The graph (fig. 248, i) shows the water
run in during the successive minutes; the shaded rectangle,
for instance, represents the amount of water (5 gallons)
GALLS s
per
M.N.
MINUTES
23456
MINUTES
Fig. 248
run in during the fifth minute. Total number of gallons
delivered = 36.
I now repeat the operation, but this time I turn the tap
on every half minute, beginning by running in \ gall, a
minute, and increasing by \ gall, each half minute. The
first delivery will be J gall., the next \ gall., and so on, the
last being 4 gall. But note (fig. 248, ii) that during the last
half of the fifth minute, when 2 gall, were delivered, the rate
of delivery was 5 gall, a minute; this column has the same
height as the corresponding column in (i), but, of course,
only half its area. The rate of flow during the half minute
was the same, though only half the 5 gall, was actually
delivered. The rate of flow during the last half of the eighth
minute was 8 gall, a minute, though only 4 gall, were de
livered. Total number of gallons delivered = 34.
THE CALCULUS: SOME FUNDAMENTALS 431
I repeat again, this time allowing the water to be in
creased every J minute, beginning by running in J gall,
a minute, and increasing by J gall, each J minute. The
first delivery is thus yg gall., the next J gall., the last f or
2 gall. Note (fig. iii) that during the last  of the fifth minute,
when 1J gall, were delivered, the rate of delivery was still
5 gall, a minute; the column has the same height as the
corresponding columns in the first two figures, but of course
only J of the area of the column in the first figure. The
8
7
G
O 2
I Z
3456
MINUTES
(in)
1234
MINUTES
(iv)
676
Fig. 249
rate of flow in the column preceding HK was the same in all
3 cases. Total number of gallons delivered (fig. 249, iii) = 33.
I repeat the operation once more, this time turning on the
tap gradually and continuously, in such a way that at the
end of the first minute the water is running at the rate of
1 gall, a minute, though only momentarily; and so on.
At the end of the eighth minute I turn off, i.e. at the very
moment when the rate of flow has reached 8 gallons a minute.
The graph (fig. 249, iv) is now a straight line, and its area is
(8 X 8) or 32 units, the number of gallons delivered.
Observe that, in all 4 figures, the rate of delivery at the
end of any particular minute is the same, for instance at
the end of the fifth minute, represented by HK; though in
432 CRAFTSMANSHIP IN MATHEMATICS
the last case, when the tap is gradually turned on, the rate
at any particular time is only momentary, since the rate is
continuously changing.
In the last figure, HK no longer bounds a rectangle, as
it did in the previous three figures; all the columns have
become indefinitely narrow. The column which HK bounded
has shrunk to a mere line which therefore cannot represent
any quantity of water delivered. Still, as it has the same
height as the series of gradually narrowing columns which
it bounded, we say that it represents a rate of flow of 5 gall.
a minute, just as the columns did. But this rate of flow
is clearly not a rate of flow during any interval of time,
however small. Hence we say it is the rate of flow at the
end of the fifth minute.
Boys ought now to understand clearly that the velocity
of a body at any instant is measured by the rate per unit
time in which distance is being traversed by the body when
in the immediate neighbourhood of that instant.
A body cannot move over any distance in no time, so that
we could not find its velocity by observing its position at
one single instant. To find its rate of motion, we must
observe the distance traversed during some interval of time
near the given instant, this interval of time being the shortest
possible. Hence the term velocity at any instant must be
regarded as an abbreviation for average velocity during a
very small interval of time, including the given instant. But
we have no means of finding such a velocity by actual experi
ment. We have to adopt other means.
It is sometimes said that acceleration at a given instant
of time is measured by the rate per unit time at which the
velocity is increasing in the immediate neighbourhood of the
given instant, or the average acceleration in a small interval
of time including the given instant.
The question, what is meant by the statement that at a
certain moment a thing is moving at the rate of so many
feet, a second ought now to be answered by all average
THE CALCULUS: SOME FUNDAMENTALS 433
pupils. Sixth Form boys should grasp the full significance
of the following formal statement: " if the magnitude
possessed by any increasing or decreasing quantity be re
presented by an areafunction, the rate of increase or decrease
of the quantity at any specified point is given by the corre
sponding ordinate function/'
Thus if any given function is regarded as an areafunction,
the corresponding ordinate function may be called the
ratefunction of the former.
Calculation of Rate functions
We may consider again the ratefunction corresponding
to the areafunction ax 3 . According to the results at the
end of the last chapter, this should be 3ax 2 .
Q X
(i)
Fig. 250
Let the curve in fig. 250 have the property that the area
under it from the y axis up to any ordinate PQ is ax 3 . How
may we determine the exact height of PQ?
Take two other ordinates (fig. ii) CD, EF, each at distance
h from PQ. Draw upon DQ, QF rectangles whose areas are
respectively equal to those of the strips under the curve
between CD and PQ, and PQ and EF. Let the curve cut
the upper ends of these rectangles in p and p'. Draw the
ordinates pq and p'q'.
(E201 29
434
CRAFTSMANSHIP IN MATHEMATICS
Although we cannot calculate PQ directly, it is easy to
calculate pq and p'q 1 '. We have:
pq X h = area CQ.
. _ area CQ
mpq h~
_ ax 9 a(x /
h
= (3* 2  3xh +
i.e. /><?  {3* 2  h(3x 
X h = area EQ.
..//__ area EQ
.<, _
a(x + /O 3
i.e. p'q' = {3* 2 + (3* + /*)}
Whatever value ^ may have, h may be taken smaller; hence
h must be smaller than 3x. Thus h(3x A) and A(3# + h)
must both be positive, and pq will necessarily be less than
PQ, and p'q' greater than PQ. By making h small enough,
we can make pq and p'q' differ from Sax 2 as little as we please.
In other words, PQ must lie between all possible positions
of pq and p'q' y and thus the value 3ax 2 is the only value left
for it to possess.
The Rate as a Slope. Here is another way of consider
ing a ratefunction. Let OQ' be the curve y = ax 3 . Let
the abscissa of any point P be x, and the abscissae of two
neighbouring points Q and Q', x h and x + h, respectively.
While x increases from x  h to x y and from x to x + A,
y increases by Q<? and q'Q', respectively. (Fig. 251.)
Hence the average rate of the latter increases must be
S*? and ?yi, i.e. tanP*X and tanPz'X, respectively.
___ PM  QN
~~ h
_ ax*  a(x  Kf
h
i.e. tan P*X = {3* 2  h(3x  h)}a.
tan
h
Q'N'  PM
 ax 3
i.e. tan
>? + h)}a,
(both as in the last example)
As h gets smaller, Q and Q' approach P, tanP/X being always
THE CALCULUS: SOME FUNDAMENTALS 435
less, and tanP^'X always greater, than Sax 2 , though by taking
h small enough, they may be made to differ as little as we
please from Sax 2 .
If PT be drawn, so that tanPTX = 3a* 2 exactly, then
PT is evidently the tangent at P. For a line through P ever
so little divergent from PT would make with the x axis an
angle greater or less than PTX, and so would cut the curve
h, is purposely exagge rated
ib make the Kdure clear
Fig. 251
in one of the possible positions of Q or of Q'. Hence PT is
the only line which meets the curve at P but does not cut it.
Thus PT holds among secants such as PQ or PQ' the
same unique position that HK holds among the rectangles
(figs. 248, 249), or that PQ holds amongst the other ordinates
(fig. 250).
In fig. 251, the slopes of PQ and PQ' measure the average
rate of change of the function during the changes of x repre
sented by NM and N'M. The slope of PT does not measure
the change during any intervals, but evidently measures what
has been defined as the rate of change of the function at the
moment (or for the value of x) represented by OM.
436 CRAFTSMANSHIP IN MATHEMATICS
Meaning of " Limit "
The common element in the three cases considered,
HK (figs. 248, 249), PQ (fig. 250), PT (fig. 251), is described
by saying that all three are examples of a limit. In all
three cases, members of a series have been brought nearer
and nearer the limit, but they have never been so near that
they could not have been brought nearer. They have always
remained " in the neighbourhood " of the limit, but in
every case the limit has been unreachable. In all three cases,
the limit is the first number outside the series.
A ratefunction is sometimes given this general definition:
Take the given function of x, and find how much its value
changes when x is raised or lowered by any positive number
h. Divide this change by h, and so obtain the average rate
of change for a change of the variable from x h to h y or
from x to x + h. The ratefunction is the limit of the quotient
and is indicated more and more closely as h gets smaller
and smaller.
The Two Main Uses of Limits
1. To define the velocity of a given point at a given moment.
If we define velocity as the quotient of a distance travelled,
by the time in which it is traversed, then " the velocity at
a given moment " is not a velocity at all.
On the other hand, if we consider the distance travelled
by the point during a series of constantly decreasing in
tervals of time, and divide each distance by the length* of the
corresponding interval, we shall again fail, as a rule, to
obtain anything that can be called the velocity of the point,
for^all the results will be different, except in the special case
of uniform motion. But if the sequence of average velocities
thus calculated follows some definite law of succession as
the interval is taken smaller, then it will generally have a
definite limit as the interval approaches zero. Thus the
limit is a perfectly definite number, associated in a perfectly
THE CALCULUS: SOME FUNDAMENTALS 437
unambiguous way, both with the given moment and with
the endless sequence of different average velocities. More
over, for small intervals of time, the average velocities are
sensibly equal to the limit, the differences being of theoretical
rather than of practical importance. It follows that although
the " velocity at the given moment " is not really a velocity
at all, it is quite the most useful number to quote in order
to describe the behaviour of the moving point while it is
in the neighbourhood of the place which it occupies at the
given moment.
2. To determine a magnitude which cannot be evaluated
directly. Consider again fig. 250. We had to determine the
height of the ordinate PQ. We found (i) that it lies between,
and is the limit of, a lower sequence consisting of ordinates
pq and an upper sequence consisting of ordinates p'q'\ (ii)
that it lies similarly between, and is the limit of, the se
quences of numbers represented by { 3# 2 h(3x  h) } a
and { 3# 2 f h(3x + h) } a\ and (iii), that the latter sequence
corresponds to the former, term by term. From these
premises it seems to be an inescapable conclusion that the
height of PQ is exactly 3## 2 , for PQ is the only line between
the two sequences of (i), and 3ax 2 is the only number between
the two sequences of (ii).
For blackboard revision work occasionally, devise questions
to emphasize these principles (the term gradient might now
be used generally):
(1) The gradient of a chord is the average gradient of the
arc.
(2) A tangent is the limiting position of a secant.
(3) The gradient of a tangent at P is the gradient of the
curve at P.
(4) The gradient of the tangent is the rate at which the
function is changing.
(5) The limiting value of the slope of a secant is the slope
of the tangent.
" In the neighbourhood of." We have spoken of the
438 CRAFTSMANSHIP IN MATHEMATICS
members of a series being " in the neighbourhood of " a
limit. What is a neighbour! That is a question of degree.
In Western Canada, a man's nearest neighbours might be
40 or 50 miles away; in an English country district, perhaps
a single mile; in a town, only a few yards; round one's own
table, only a few inches. So with number sequences: it is
just a question of degree. For instance, we know that TT comes
within the interval 31 and 32, and therefore 31 and 32
are neighbours of TT. But TT also comes within the smaller
interval 313 and 315, which are therefore closer neigh
bours of 77. Again, TT comes within the interval 3*1414
and 3*1416, which are therefore still closer neighbours of TT.
And so we might go on. However close our selected neigh
bours of 77, we can always find still closer neighbours. Thus
77 always has neighbours no matter how small the interval
in which he is enclosed. It is all a question of standard of
approximation. The important thing, when dealing with
limits, is that we must never think of the interval shrinking
to nothing. Think of the interval as always large enough for
standing room both for 77 itself and some neighbours. The
neighbours cannot be thought of as disappearing altogether.
Secant to Tangent Again
The gradient of a straightline graph AB is determined
easily enough: it is the ratio, ordinate j> /abscissa x. The
ratio may be determined
from any selected bit of
the line. Or, if we like,
we may increase the line,
say to BC,and take the ratio
CD ( increment of y) to
BD ( increment of x).
Fig . 2S2 But if the graph is a
curve, the gradient at any
specified point on the curve is determined by the tangent
at that point. A ruler held against the edge of an ordinary
THE CALCULUS: SOME FUNDAMENTALS 439
dish is, practically, a tangent at a point on the ellipse. If,
then, we want to determine the gradient of a curve, why
not just draw the tangent and measure the angle it makes
with the x axis?
With a circle this would be easy enough: we should draw
a radius to the point and then a line at right angles; and
there are simple rules for certain other curves. But merely
to hold a ruler against a curve, and to draw a line, is not to
draw a tangent that we can accept. Circulate amongst the
class copies of a mechanically drawn parabola, tell the boys
to draw a tangent at the point P, and then to measure the
angle that the tangent makes with the x axis. The angles
will probably be all different. Clearly the method will not
do, for the angles ought to be the same in all cases.
If we draw a secant instead of a tangent, and find the
gradient of the secant, we shall evidently have the average
gradient of the curve between the two points P and Q where
the secant intercepts the curve. Would that help?
Yes, but if the points are
far apart, as P and Q x , the
slope of the secant, and
therefore the average gra
dient of the curve between
the two points, differs much
from the gradient of the
tangent PT. If we bring
the points closer together,
say P and Q 2 , the gradient
of the secant is nearer the
gradient of the tangent. If p .
we bring Q down to Q 3 the
gradient of the secant PQ X is still nearer the gradient of the
tangent. It is this gradient of the tangent that we have to
find somehow.
[Some teachers prefer that numerical considerations like
those that follow should precede the more general work
440
CRAFTSMANSHIP  IN MATHEMATICS
concerning the graph, as outlined in the earlier part of this
chapter. I have seen equally satisfactory final results obtained
from both sequences.]
Let us actually calculate the gradients of successive
secants, and see if we can learn anything from the results.
On a parabola we will select a point P where x = 1 (and
.". y = I 2 1), and keep this fixed. We will also place a
point Q on the curve, at first where x = 15 (and .'. y = (1*5) 2
= 225). Thus, since P is (1, 1) and Q is (15, 225), the
increment of x is '5, and the increment of y is 1'25. (The
piece of line PQ may be looked upon as an " increase "
of the line AP; hence the term " increment " may usefully
be applied to the corresponding increases of x and y.)
The gradient of the secant = = == 2*5. Now
PV '5
125
THE CALCULUS: SOME FUNDAMENTALS 441
bring Q gradually closer to P. Let the next four x values be
14, 1*3, 12, 11; then the corresponding y values are (1*4) 2 ,
(1*3) 2 , (12) 2 , (1*1) 2 . The gradient calculations may be sum
marized thus (the x and y increments are often indicated by
h and k t respectively):
* = ON =
15
14
13
' 12
11
y = x * = ON 2 = NQ =
225
196
169
144
44
2
121
* = QV
h = PV = MN =
125
5
96
4
69
3
21
1
21
Gradient  J = ^  =
h MN
25
24
23
22
Note how the value of the gradient has diminished from
25 to 21. We cannot write h = 0, or the denominator of
our ratio would equal 0, and the ratio would have no
meaning. But we may continue to diminish the values of
the x increment, and calculate the gradient as before. We
may make the increments as small as we please. Let us
calculate the gradient when the successive values of x for Q
are 101, 1001, 10001, 100001, so that the x increments
are 01, 001, 0001, 00001. We cannot draw the figure,
for the increments are much too small to be shown.
* = ON
101
1001
10001
100001
y = x 2 = ON 2 = NQ =
10201
1002001
100020001
10000200001
fc = QV =
h = PV  MN =
0201
01
002001
001
00020001
0001
0000200001
00001
Gradient  \ = J  =
h MN
201
2001
20001
200001
442 CRAFTSMANSHIP IN MATHEMATICS
We observe (1) that however small we make h (the in
crement of x), the value of the gradient always exceeds 2;
(2) that the smaller we make the increment, the smaller is
the excess of the gradient over 2. Evidently we can approach
to within any degree of approximation we like to name; it
is only a question of making h small enough to start with.
We observe also that the more nearly the value of the gradient
approaches 2, the more nearly does the secant approach
the position of the tangent. As long as the secant remains
a secant, it can never be a tangent, and it must always have
a gradient in excess of 2. But the successive gradients seem
to compel us to infer that the gradient of the tangent itself,
and therefore of the curve, is 2. Thus we regard 2 as the
limiting value of the gradient of all possible secants. It is a
value that is never quite reached by any secant, for the tangent
stands alone, outside them all, foursquare and defiant!
Thus we have found that, for the function y = x 2 , the
gradient of the point P, where x 1, is 2.
We may arrive at the same result by arguing more generally,
merely calling the increments, h and k.
The coordinates of P are (1, 1).
The coordinates of Q are (1 + h> 1 + &)
Since y = x 2 ,
(1 + *) = (1 + h?
:. k = 2h + h 2 ,
QV k 2h

From this point on, argument nowadays commonly
proceeds thus:
As Q approaches P, so h tends towards 0. We have to
k 2/z 4 h 2
find the limit to which  or ' tends as h tends towards
h h
9/j _1_ /2
As long as h is 4= 0, Ln \ n = 2 + h, and as h+
THE CALCULUS: SOME FUNDAMENTALS 443
2 + h + 2. If we decide that 2 + h must differ from 2 by
less than 1/1000000, there is no difficulty; we merely give
to h a value less than that, e.g. 1/1000001.
k
In the limit, as h > 0, T > 2.
A
Hence the gradient of the curve at P = 2.
We may now find the gradient at any point P (x y y).
The coordinates of Q are (x + A, j> + A).
.' (y + *) = (* + A) 2
= * 2 + 2xh + A 2 ,
/. k = 2xh + A 2 .
Hence the gradient of PQ = ^ = 2x + h if h 4= 0.
A
Now as Q approaches P, A > 0.
/. the gradient of the curve at P limit of (2x\h) as A> 0,
I am not quite happy about the language of this argument,
though it is now in common use and has been designed to
get over the old difficulty of infinitesimals and of the absurdity
of dividing by 0. But even able boys in the Sixth sometimes
admit that the reasoning is not clear to them, saying that
they feel they take a leap over the final gap to the limit.
The teacher must insist that the gap is really never crossed,
that the interval still remains, that the limit is always there
with a crowd of neighbours who vainly strive to reach him;
that every neighbour has a value a little greater than 2x (or,
in some of our earlier illustrations, a little less), and that the
value 2x is a solitary value, which therefore we feel bound to
assume is the value which belongs to the Limit, and to the
Limit alone.
Revise: The function y = x 2 . To calculate the ordinate
for any value of x, work out the value of x 2 . To calculate the
gradient for any value of x, work out the value of 2x.
444
CRAFTSMANSHIP IN MATHEMATICS
Thus x 2 may be described as the formula for the ordinate,
and 2x as the formula for the gradient. In other words, the
function x 2 gives the ordinate, and the function 2x the gradient,
for any value of x.
x 2 is the original function which defines the curve; 2x
is called the derived function of x 2 . The process of finding
the derived function of a given function is called differentiation.
Since the gradient of the tangent to y x 2 at any point
P is 2#, the gradient where x = 1, is 2; where x = 2, is 4;
where x = 3, is 6; &c. Does this square with the work we
have done in pure geometry? We found (p. 409),
j^r^ ^ r u 1 i latus rectum
gradient of tangent to axis of parabola =
5 * F
or gradient of tangent to tangent at vertex =
ordinate
ordinate
latus rectum'
Let the tangent at the vertex be the x axis, and let the axis
of the parabola be the y axis. Let S be the focus, and let
the latus rectum LSP be unit length.
THE CALCULUS: SOME FUNDAMENTALS 445
Half the latus rectum = SP = . Since PN = half the
distance of P from the directrix (not shown), PN = PS = J.
Hence the coordinates of P are (, ).
At the point Q (1, 1), gradient of tangent to OX
ordinate __ 1 _ ~
half latus rectum
At the point W (2, 4), gradient of tangent to OX
ordinate _ 2 __ .
half latus rectum \
At a point Z (3, 9), gradient of tangent to OX
ordinate 3 ,
=  = o.
half latus rectum \
Clearly then, the new method of finding the slope of the
tangent does produce a result absolutely accurate, not merely
approximately accurate. Evidently the " limit " argument is
sound, though we must always remember that the limit
is outside the sequence under consideration, never reached
by any member of the sequence.
The Calculus Notation
We have used the letters h and k to denote the increases
(" increments ") in the values of x and y. But the increments
always actually considered are very small, and the symbol
generally used to denote them is the Greek letter delta
(A or 8) prefixed to the value of x or y from which the in
crement begins. Pronounce A* as " delta x "; the symbol
A# must be taken as a whole; A is not a multiplier and has
no meaning apart from the x and y to which it refers.
Remember, then, to write Ax instead of /r, and by instead
of*.
446 CRAFTSMANSHIP IN MATHEMATICS
A# means " the increment of x "; Ay means " the in
crement of y ".
Ay  . increment of y rp, A , ,
means the ratio = . 1 he A s cannot be
A# increment or x
cancelled.
Treat  exactly as if written ; it measures the average
A# h
gradient of the graph over the interval between x and
x + A#.
The limit of ^ is the gradient of the graph at the point
. dy
given by x. It is sometimes written D(y), sometimes ^.
j ax
But the curious thing is that, although ~ looks like a ratio
dx
or a fraction, it is not a ratio or fraction. The symbols dy,
dx, written separately, have no meaning. The limit of A#
is not dx\ the limit of Ay is not dy. ~ is just a single symbol.
dx
~ is always a ratio of real value; f is not a ratio at all
A# dx
and is therefore very misleading to the eye.
The process of finding D(y) or is called differentiation.
, ax
D(y) or f has received various names:
dx
(1) The derivative of y or f(x) with respect to x.
(2) The differential coefficient of y with respect to x.
(3) The derived function of y with respect to x.
We will differentiate x*. Let y = re 4 . When x is increased
to x + A#, let jy be increased to y + Ay. Then:
y + Ay = (x + A*) 4
= * 4 f 4* 3 A* f 6* 2 (A*) a + 4*(A*) 8 + (A*) 4 ;
f 6* 2 (A*) a 4 4*(A*) 3 f (A*) 4 ,
THE CALCULUS: SOME FUNDAMENTALS 447
Hence as A* > 0,  ~ > ky?.
A*
.'. ^ = 4*'.
dx
Let the class discuss the result (or one like it) critically. In
particular, discuss the significance of the arrows. Forms
and language that might pass muster in an examination room
should be subjected to the closest scrutiny in class. It is a
fact that those boys who have acquired facility in working
out the ordinary stock exercises in the calculus are often
nonplussed when crossexamined in the underlying funda
mental notions.
There can be no doubt that the idea of derived functions
is best introduced as a generalization of the familiar
ideas of connexions between area functions and ordinate
functions, ordinate functions and gradient functions, &c.
The notation f should not be introduced too soon. D(y)
ax j
is much preferable. The symbol ~ originated with Leibniz
(not with Newton), and it expresses a view of the nature of a
differential coefficient that is out of harmony with modern
ideas and conflicts with the doctrine of limits. Originally
the view was that any finite value of the variables y and x
is really the sum of a vast number of " infinitesimal " values
which, though immeasurably small, have yet a definite
magnitude. Thus the differential coefficient f was looked
dx
upon as simply the ratio of the " infinitesimals " of
two variables, the ratio being finite and measurable (much as
the weights of atoms are measurable), in spite of the smallness
of the terms. This view is no longer held. The expression
^ is not a ratio at all but only the limit which the ratio
dx
of the increases of the variables approaches as the increment
of x approaches zero. Naturally the learner is greatly puzzled
4:48 CRAFTSMANSHIP IN MATHEMATICS
if he is told to write the derivative in the form of a
fraction and is then forbidden to think of it as a fraction.
Thus it is much the best plan to withhold the Leibnizian
notation at first. Use the symbol D(y) instead; this symbol
reminds the pupil that he is seeking a function which he is
to derive from the given function y by means of a definite
rule of procedure. This relationship between functions is
the essence of the whole matter.
Integration. Like current ideas about the nature of a
differential coefficient, those about the nature of an integral
also show traces of the erroneous mathematical philosophy
of earlier days. The problems first systematically studied by
Wallis came to be regarded as having for their aim the
summation of an " infinite " number of " infinitesimals " dy,
of the form y.dx. This view is still represented not only by
the usual notation I = \y.dx, which (like dy/dx) was in
troduced by Leibniz, but also by the common statement
that an integral is the sum of an infinite number of infinitely
small magnitudes. With the rejection of the notion of an
infinitesimal as a definite atomic magnitude, this statement,
and the notation which expresses it, have become inadmissible.
If dx has any numerical significance at all, it stands for the
increment h when h is zero. Hence the product y.dx is also
zero for all values of y, and therefore the summation repre
sented by jy . dx is the summation of a series of zeros! I is
not the sum of an infinite number of products; * it is
simply the limit of the sum of a finite number of pro
ducts.
There is neither need nor warrant for introducing the
term " infinite " at any point of the discussion. If we sub
stitute the useful A# for the absurd dx y we may still usefully
retain the Leibnizian mode of expression I = fy . A#, but the
symbol " / " must now be read, " limit of the sum as A#
approaches zero ".
THE CALCULUS: SOME FUNDAMENTALS 449
Interpretation of .
n Ay distance j j A
1. r^ = : = average speed during A#;
A* time * F fe
dy limit of average speed = " instantaneous "
d~x ~~ speed.
2.  average slope of curve during interval A#;
L\X
j = limit of average slope = limit at point P.
The two problems (1) to determine the rate of increase
of a function and (2) to draw a tangent to a curve, are really
identical; if we have a general method of determining the
rate of increase of a function f(x) of a variable x, we are
able to determine the slope of the tangent at any point (x, y)
on the curve.
Points for emphasis. We will once more stress the points
to be kept in the forefront of the teaching.
The pupils should be told plainly that the old idea of
infinitely small quantities has been definitely abandoned.
The real explanation of the whole thing was first put forward
by a German mathematician, Weierstrass, about the middle
of the nineteenth century. The subject had been sound
enough; so, virtually, had been the mathematical procedure,
but the explanation had been wrong.
The problem was to retain an interval of length A, over
which to calculate the average increase, and at the same time
to treat h as if it were zero. As Professor Whitehead says,
" As long as we look upon ' h tending to a ' as a fundamental
idea, we are in the clutches of the infinitely small, for we
imply the notion of h being infinitely near to a. This is
what we want to get rid of." " The limit of f(h) at a is a
property of the neighbourhood of a." " In finding the limit
(E291) 30
450 CRAFTSMANSHIP IN MATHEMATICS
Z/Ov I Jj\
of ^ 1 at the value of the argument h y the value
h
(if any) of the function at h = is excluded. But for all values
of h except h = we can divide through by h." " In the
neighbourhood of the value for h, 2x + h approximates
to 2x within every standard of approximation, and there
fore 2x is the limit of 2x + h at h = 0. Hence, at the value
for h, 2x is the limit of (* + h Y ~ **
h
The difficulty of former mathematicians was that on the
one hand they had to use an interval h over which to calculate
the average increase, and on the other hand they wanted to
put h = 0. " Thus they seemed to land themselves with
the notion of an existent interval of zero size." Presentday
mathematicians avoid that difficulty by using the notion
that, corresponding to any and every possible standard
of approximation, there is still some interval.
My own experience is that when Sixth Form boys are
puzzled over this question, their puzzlement is almost always
due to the fact that they have got hold of the term infinity,
and do not understand what the term signifies.
Books to consult:
1. The Teaching of Algebra , Nunn.
2. An Elementary Treatise on the Calculus, Gibson.
3. Course of Pure Mathematics, Hardy.
4. Applied Calculus, Bisacre. (An outstanding book.)
WAVE MOTION 461
CHAPTER XXXI
Wave Motion: Harmonic Analysis:
Towards Fourier
Sine and Cosine Curves. Composition
The pupils will, of course, be thoroughly familiar by
this time with the radian notation, and will understand that
the reason for measuring angles in radians is that theoretical
arguments are simplified. They will know that TT radians= 180;
that as an angle of 9 radians is subtended by an arc of Or,
the length of an arc of a circle = r9\ that the symbols 6 and
(/> are commonly used in circular measure, and the symbols
a, /?, y for measurements in degrees. They ought also to
know that, when an angle is small, its circular measure may,
in approximation calculations, be substituted for its sine
(or tangent); and that, when it is not small, the values of
the sine and cosine may still be expressed approximately in
circular measure by means of the simple formulae sin 9 =
3 9 2
, cos0 = 1 . The proofs of these may be given at
6 2
an appropriate stage, but a simple graphic method is easily
devised to suggest that the formulae are approximately true.
When boys are first introduced to angles greater than
360, they are inclined to doubt if they are dealing with real
things, and to be a little sceptical about the practical value
of the work in hand. Light, however, begins to dawn when
they are introduced to Simple Harmonic Motion, to waves,
and to spirals.
They must be made to understand clearly that the values
of the ratios connected with an angle are repeated endlessly
in cycles as the angle rises.
They will, of course, be thoroughly familiar with the sine
and cosine curves. With very little practice they can make
452
CRAFTSMANSHIP IN MATHEMATICS
a supply of these curves for themselves by running them
off from Fletcher's trolley arranged for uniform motion;
with care, these curves may be obtained to a surprising
degree of accuracy. Draw tangents to the succession of
crests, then the axis midway between them. The chief
ordinates are the perpendiculars at those points of the axis
midway between the nodes.
Point out that all sine curves have the same general
shape and properties; that smx gives a wave curve of period
Y
2?r with successive maximum and minimum values at +1
and 1 respectively; that sinpx gives a wave curve of period
o o
; that asin(px\e) gives a wave curve of period ,
P P
with successive maximum and minimum values of \a and
a, respectively, the effect of e being merely to displace
the curve along the axis. These fundamentals must be
mastered. The p y the a y and the e are veritable traps for
the unwary beginner; the inner significance of the three
symbols should be expounded and emphasized again and
again.
From his earlier knowledge of graphs, a boy may, without
further instruction, graph one or two easy cases of compound
periodic functions. We give two examples adapted from
examples in Siddons and Hughes' Trigonometry, the first,
2 sin* + 3 cos#, consisting of two periodic functions of
the same period (fig. 256), and the second sin3# + 2 sin*,
periodic functions of different periods (fig. 257). In each
case the two functions are first plotted separately (the curves
are shown by lighter lines), then the required composite
WAVE MOTION
453
curve is obtained by means of points determined by taking
the algebraic sum of the ordinates of the constituent curves.
Y
4
1'ig. 256
For instance, in the second case pm = pn + pq. Note in
the first case where we are compounding functions of the
Y
\
^
Fig. 257
period, the result is a 'e curve; in the second case,
where the functions to be compounded are not of the same
454 CRAFTSMANSHIP IN MATHEMATICS
period, the result, though a periodic curve, is not a sine
curve. If in the second case we slide the half curve TT to 2ir
along the x axis up to the y axis, the upper and lower halves
will easily be seen to be symmetrical. Observe that in the
case of sin 3x the period is  = 120.
<j
In all such cases the shape of tlje composite curve can
be obtained only by plotting a wide range of ordinate values,
though this is always simply done by algebraic addition,
and a pair of dividers will soon give the necessary number
of points. The curves are a little tricky to draw, because
of the minus quantities to be added.
This kind of exercise need take but little time. I have
known boys work through half a dozen in an hour. The
general shapes of the sine and cosine curves are already
familiar, and as the axis can be divided up and the principal
ordinate put in at once, the constituent curves can be sketched
in in less than a minute. It is assumed, of course, that the
significance of p, a, and e has been fully grasped. But the
negative quantities to be considered when building up the
composite curve nearly always give trouble. No calculations
are, however, necessary. Let the dividers do the work,
unless, in some very exceptional instances, rigorous accuracy
is wanted. The general form and what it teaches is the
main thing.
Waves and their Production
Since, in most instances, waves are periodic phenomena,
they afford excellent concrete examples of periodic functions.
No one can appreciate the most striking triumphs of physical
science who has not given some attention to the mathematics
of wave motion. In fact, wave motion now forms the very
basis of the study of the greater part of physics, and, after
all, the necessary mathematics of the subject is, in all its
main factors, quite simple. It need hardly be said that
that new and rather formidable subject, Wave Mechanics,
is outside the scope of school practice.
WAVE MOTION 455
The teaching of such characteristics of waves as resistance,
persistence, and overshooting the mark, is part of the business
of the physics master. The mathematical master is concerned
mainly with considerations of the form of the wave and its
analysis. Let beginners first read through Fleming's Waves
and Ripples, and so supplement the work they have already
done in the physics laboratory; the mathematics will then
give them little trouble. But, if they begin the mathematics
of wave motion before they have acquired in the laboratory
a considerable amount of practical knowledge of the subject,
they will never be quite sure of their ground.
The boys will probably be familiar with the device of
producing a train of waves by means of a length of narrow
stair carpet, or a sandfilled length of rubber tubing, or a
length of heavy rope: these things are part of the stock
in trade for teaching wave motion in the physics laboratory.
An instructive experiment is the following: take a common
blindroller about 5' long, with a pulley runner fixed at each
end. Into the roller drive 37 4" nails, at 1J" intervals, in
the form of a uniform spiral of 3 complete turns. The nails
should be separated from one another by a uniform interval
of 30, so that the 1st, 13th, 25th, and 37th are in the same
straight line; the 2nd, 14th, and 26th in another straight
line; and so on. Support the roller in a horizontal position
in front of a white screen, and turn it by means of an im
provised crank. Let a distant light throw on the screen
a shadow of the rotating roller. Observe how the shadows
of the nailheads exhibit progressive wave motion. Observe
the movement of any one particular shadow; it is an example
of simple harmonic motion (see Chapter XXXVII). The
travelling shadowwave, constituted by equal simple harmonic
motions of the shadows of the nailheads, is a progressive
harmonic wave. The shadow of any head differs in phase
from that of its neighbours by a constant amount of 30.
Note that each nail remains in its own vertical plane; the
progressive horizontal movement is one of form only. The
boys must distinguish between (1) the actual to and fro
456
CRAFTSMANSHIP IN MATHEMATICS
movements of elements in a wavemedium, and (2) the move
ment of the wave itself. The second is merely an appearance,
resulting from the successive movements of the first. The
first has the effect of making successive sections of the medium
(as we may conveniently call it) assume one after another
the same shape. The shape therefore appears to be some
thing moving along.
The waves on the surface of the sea, away from the shore,
are good examples of progressive waves. If their outline
were exactly a sine curve, as theoretically it should be, we
should have an example of a harmonic progressive wave.
Common Wave Formulae
In figure 258,
the wavelength L t L 2 = L 2 L 3 = A,
the amplitude = PQ = a.
I x
Fig. 258
If T is the periodic time of the wave (i.e. time to complete
a vibration), it follows from first principles that v = A/T = A,
where v = velocity and n frequency.
Let Lj# be d\ let pq be h. Since the curve is a sine curve,
A = 360 = 2?r. Hence the number of degrees in d = X d.
277 A
For T write p. Then, wherever q is taken along L^Lo, it
A
may be found, by actual measurement, that h = a sind.
This is a fundamental formula.
Let y = a sinpx describe the wave outline in a given
position, x being measured from Lj. If the curve move to the
WAVE MOTION 457
right with a velocity v, its form after t seconds is given by
the formula,
y a s'mp(x vt) (i)
If to the left with the same velocity,
y = a sinp(x + vt) (ii)
These formulae are simply applications of the general prin
ciple that if a graph is moved a distance d parallel to the
x axis, (x d) must be substituted for x in the formula.
Since p , (i) and (ii) may be expressed thus:
A
y = a sin (x vt).
A
The actual significance, in the graph, of each symbol in
this formula must be understood.
Compound Harmonic Waves
Let two boys near each other on the edge of a pond,
or other suitable sheet of still water, each produce a series
of waves by striking the water rhythmically with a stick.
Let the frequency of the blows be 2 to 3 (say 2 in 2 seconds
and 3 in 2 seconds, easily done after a little practice with
watch in hand), and suppose the waves to travel with the
velocity v. A pattern will result from the five waves which
are produced every two seconds, and it will be regularly
repeated, though gradually fading away into ripples. But
this pattern will no longer represent simple harmonic waves,
for the shape which appears to move along the water beyond
the ends of the line joining the centres of disturbance, is
no longer a simple sine wave; the length, the frequency,
and the amplitude of the resultant waves will be different
from the length, the frequency, and the amplitude of the
component waves. At points reached simultaneously by
crests and troughs belonging to the component wavetrains,
458 CRAFTSMANSHIP IN MATHEMATICS
the elevation or depression of the surface is exaggerated.
All this should be confirmed by observation.
To calculate the resultant disturbance due to the two
component waves (assumed to be simple harmonic waves),
we adopt the principle, which accords with observation,
that the actual displacement at any point is equal to the
algebraic sum of the displacements due to the waves separately.
If the first wavetrain existed alone, the displacements
produced would be represented by moving the curve
y = a : sin x with velocity v towards the right. If the
^i
second wavetrain existed alone, the displacement produced
would be represented by moving the curve y~a 2 s'm (x c)
2
with velocity v towards the right. Here, c is the x coordinate
(at t = 0) of the nearest point of the wave from the centre
of disturbance, comparable with L in fig. 258.
Thus the actual character of the resultant composite
wave is represented by the graph
. 2ru , . 27U, x
y = a l sin x j a 2 sin (x c),
AJ X 2
moving to the right with a velocity v.
This evaluation from first principles is really very simple,
but, unless it is associated with at least a little experimental
work, it may prove difficult for average boys. The formula
is a key formula and should be mastered.
Had there been three boys at the pond side, each pro
ducing waves by striking the water rhythmically, all the
waves being of different length, the composite waves would
have been more complex, and the necessary formula for the
graph would have consisted of three terms. So generally.
WAVE MOTION 459
Comparison of Periodic and non Periodic
Functions
Functions of the form s'mpx and cospx (where p = ~\
\ A /
have much the same relation to periodic curves as x has to
nonperiodic curves. The simplest nonperiodic curve is
the straight line y Ax (we write it in different forms,
according to circumstances; e.g. y mx f c); and the
simplest periodic curve is y = A s'mpx (also written in
different forms according to circumstances).
With our former nonperiodic work we soon learnt that
the curve y = A I( * +A 2 # 2 was more complex than y = Ax,
and that y A^x + A 2 x 2 + A 3 x? was more complex still;
and so on; a quadratic function was more complex than
a straightline function, a cubic more complex than a
quadratic. Still, however complex the function, it was
always a question of the addition of a number of terms;
the actual graphing was simple enough though tedious if
the terms were many.
So it is with periodic functions, where the curve, whether
simple or complex, recurs endlessly, and makes a continuous
wave. The effect of adding to y = A s'mpx the term A 2 sin 2px
may be compared with that of adding to y A* the term
A 2 # 2 ; in each case we obtain a form of greater complexity.
By adding further terms we get, in each case, still further
complexity, save that in the former case the successive curves
all have the period A or a submultiple of this.
P
The standard form of a periodic function may be written:
y Aj s'mpx + A 2 s'mSpx + A 3 sin3j># + . . . f A r s'mrpx.
It was the French mathematician Fourier who first observed
that a periodic function of unlimited complexity may be
described by a formula of this type. The process of deter
mining the components of which a given periodic function
460
CRAFTSMANSHIP IN MATHEMATICS
is the resultant is known as harmonic analysis. Fourier's
statement is known as Fourier's theorem.
Let the boys consider an illustration of this kind. Let
them inagine a water wave sent out with a velocity v, of
length A, and frequency 1 per second; the wave would form
a simple sine curve, such as we see on any disturbed water
surface. Now let them imagine a second wave, sent out from
the same point, independently but at the same moment, at
a frequency of 2 per second, with the same velocity v and
therefore of a wavelength A/2. This second wave would
not have the appearance of its independent self but would
be imposed on the other, and what we should see travelling
along the water surface would be a composite wave. Now
let them imagine a third wave to be sent out from the same
point, independently but at the same moment as the other
two, at a frequency of 3 per second, with the same velocity
v, and therefore of a wavelength A/3. (Remember that
v = nX y always.) This third wave will not, any more than
the second, show itself independently; it will simply make
the previous composite wave still more complex. And so
we might go on. The waves sent out independently might
be shown thus:
[, \ (
/
ot
is
/
s
M
s
s
S
/
s
/
Zir
k
/
x i 
/
"\
/
\
/
\
M
/
\
/
s
s
2.
\
/
\
/
Zir
\
/
X
Fig. 259
WAVE MOTION
461
All 3 waves start from O, and since they travel with the
same velocity they reach M at the same moment, but by
that time the first will have completed 1 of its periods, the
second 2, the third 3. Imagine a whole series of waves sent
out in this way, each of them with a wavelength which is
a submultiple of A, though not necessarily all the members
of the sequence A, A/2, A/3, A/4, &c.: some of the series may
be missing. Now imagine the water to be suddenly frozen,
so that the wave would be set in ice, and its section readily
drawn. We might have a composite wave like fig. 260,
OM composing a unit which would be repeated endlessly
until the wave died away in a ripple. The problem is, how
/
\
^\
/
\
r\ .
V
y
\^
/
\
J
\J
^ x ^
Fig. 260
are we to analyse this curve, in order to discover all the
simple curves of which it is compounded.
The waves need not all have been sent out with the same
amplitude (<z), as shown in fig. 259. Neither need they have
been sent out from exactly the same point; one might have
been started 40 or 50 farther along the axis than the others.
And remember that a cosine curve is produced from a sine
curve merely by pushing it forwards or backwards 90 along
the axis. We may therefore easily find cosines as well as
sines in our formulae; it is all a question of convenience,
depending on the particular curves under consideration.
Briefly, Fourier's statement was this: Any repeated
complex wave pattern of length A may be produced by adding
to a certain fundamental sine or cosine curve of length A,
sine or cosine curves of the proper amplitudes whose lengths
are A/2, A/3, A/4, &c. Conversely, the complex pattern may
be revolved into its original component sine and cosine
462 CRAFTSMANSHIP IN MATHEMATICS
curves, since any of the unknown amplitudes may be deter
mined at will.
Observe that the main difficulty in analysing the complex
curve arises from the fact that the component curve may be
of different amplitudes. The general expression for the
complex curve may be written in different ways, though
they all mean the same thing.
(i) y = a + (#1 sin px + b l cos px)
+ (a 2 sin2/># + b 2 cos2/>#) + . . .
(ii) y = a + a sin(* + i) + # 2 sin(2# + a 2 )
+ a 3 sin(3# + a 3 )+ . . .
(iii) y = a + a sin(0 + a) + b sin(20 + j3)
+c sin(30
Remember that p = 2?7/A.
The constant a meets the case in which the x axis is
not identical with the common axis of the various harmonic
curves.
Observe that each term in the above expressions represents
a simple harmonic function. Those harmonics in which the
coefficient of x is an odd number are called odd harmonics;
those in which the coefficient of x is even are called even
harmonics. Observe, too, that the second term gives a curve
with twice as many complete waves, the third term a curve
with three times as many complete waves (and so on), as the
first or fundamental term. This is exemplified in fig. 259,
where the " period " of the second term is \ the period of
the first, the period of the third is \ the period of the first;
and so on. The frequencies are therefore twice, three' times,
&c., the frequency of the first.
Curve Composition
Let us first consider curve composition. It is very simple
Plot to the same axis the successive components, a l sin(#+ai)
a z sin(2# + a 2 ), &c., and then add the corresponding ordinates
to obtain the respective ordinates of the composite curve
WAVE MOTION
463
(cf. figs. 256, 257). Since the first or fundamental term is
represented by the period to 2?r, the wave will consist of
repetitions of the first portion between x = and x = 360.
We give the graph of
100 sin* + 50 sin(3* 40),
from to 360. The first or fundamental term, 100 sin x,
represents the first harmonic with an amplitude 100 (=a l in
100
50.
Fig. 261
general formula). The second term, 50 sin(3# 40), is
the third harmonic with an amplitude of 50 (=a 2 ), and
consists of three complete waves within the period of the
first.
The function consists of only odd harmonics, and in
virtue of this fact the graph possesses a special kind of
symmetry characteristic of all curves containing only odd
harmonics. If the portion of the graph from n to 2?r be made
464
CRAFTSMANSHIP IN MATHEMATICS
to slide to the left, to the position to TT, it will be the re
flected image of the half above the axis. (Cf. fig. 257.) Note
that the composite curve is not a sine curve.
Had the function contained the absolute term a , say
y = 70 + 100 sin x + 50 sin(3*  40),
the graph would be the same as before but raised vertically
70 units. The line of symmetry referred to above would then
no longer be the x axis.
We give a second example, this time consisting of the
first and second harmonics:
y = 10 roi(6 + 30) + 5 sin(20 + 45).
We will plot the graph from a tabulated series of values,
though this is really unnecessary.
If 10sin( + 30)= < y 1 , and 5 sin (20 + 45) = y 2 , then
when = 0, yi = 10 sin 30 = 5, and y 2 = 5 sin 45 =
i
also when 6 = 30, y l = 10 sin 60 = 866 and y 2 = 5 sin 105
= 5 sin 75 = 48. Similarly other values may be calculated.
Note the device of running off horizontals from a graduated
circle. Since the " period " in the x axis is divided into 12
equal parts, we divide the circumference of the circle also
into 12 equal parts.
WAVE MOTION
465
Values ^
of 6 f
0.
30.
60.
90.
120.
y\ =
5
865
10
865
6
y2 =
35
48
13
35
48
yi + y* =
85
1345
113
515
02
Observe that as the point on the smaller circle rotates at
twice the rate of a point on the larger, it is only necessary to
divide the smaller circle into half as many parts as the larger.
Set up, say, 12 ordinates for the whole line to 360, then
divide the circumference of the larger circle also into 12
parts, and run off parallels to cut the ordinates. Each second
harmonic will embrace only 6 of the 12 ordinates, and hence
only 6 parallels from the smaller circle are required. The
radii of the circles are, of course, equal to the amplitudes
of the respective harmonics. Observe the plan for fixing
the first point of each harmonic.
Functions with more terms than two are treated in exactly
the same way, but naturally the composition is a tedious
operation.
Curve Analysis
Secondly, we come to the decomposition or analysis of
a composite curve. This is much less simple than the reverse
operation.
The composite curve may be the resultant of two or more,
perhaps a large number, of harmonics. But it does not at
all follow that, because a particular harmonic, say the ninth,
has been included in the building up, therefore all the earlier
ones (in this case the first 8) of the series are included too.
How are we to discover which harmonics are included, and
how are we to draw them?
Whatever scheme we adopt, it is advisable, when we have
discovered the component harmonics, to draw them all
carefully, to compound them again, and to see if the result
corresponds to the original curve.
(291) 31
466
CRAFTSMANSHIP IN MATHEMATICS
Let integration wait until a later stage. Let the boys
first learn what the new thing is really about. Let them
consider the few simple cases which may easily be solved
graphically, and after all, these are the cases of greatest
practical importance (for instance those that are concerned
with the theory of alternating currents). Such cases may,
with sufficient approximation, be represented by the sum of
two or three harmonic terms.
We select as an example one of Mr. Frank Castle's
engineering problems. The curve in the figure is drawn
'%
^
1*
I
vz
j^" ^^Sk
_jr IV
7 [\
O 1 2 3 4\
y_
6
7
G
9
O \\ 712
\
/
>
\
V
N
\
/
through 12 successive positions of a slide valve, corresponding
to intervals of 30 of the crank, beginning at the inner dead
point. It is required to analyse the motion so as to express,
in the form of a series of harmonics, the displacement of the
valve from its mean position. (Practical Mathematics y p. 459.)
Were the curve divided more symmetrically by the x
axis, we should suspect comparatively little deviation from
the first harmonic, i.e. an ordinary sine curve. But, fairly
obviously, it is compounded with other harmonics as well.
Run off the lengths of the ordinates to the edge of a
paper strip as shown in fig. 263. Use the strip for plotting
the points in fig. 264, but first reverse it, so that point 8
is at the top and point 2 at the bottom.
WAVE MOTION
467
For the first harmonic. Let of the strip coincide with O
in fig. 264, and mark off these distances: to 6 on the ordinate
through O, 1 to 7 on the ordinate through 1, 2 to 8 on the
ordinate through 2, 3 to 9 on 3, 4 to 10 on 4, 5 to 11 on 5 (six
measurements in all). Observe that these distances on the
Fig. 264
strip give, successively, y Q  y^ y 1 y 7 , y 2 j; 8 , y 3 y g ,
y ~~ Jio> Vb ~~" Vii Draw a curve through the points, then
the second half of the curve, below, through points obtained
from the same measurements, reversed. Draw a tangent to
this curve at a maximum or minimum point, HK, MN.
The amplitude a' is half the distance from the tangent to
the axis; it is 7/2 = 35.
468
CRAFTSMANSHIP IN MATHEMATICS
The magnitude of the angle a can be obtained by measuring
the length Op. The distance O to 6 = 180; hence Op = 1518.
Thus a = 180  1518  282.
For the second harmonic. The successive distances for
the ordinates to be taken from the strip are (0 to 3) + (6 to 9),
(1 to 4) + (7 to 10), (2 to 5) + (8 to 11). Observe that these
distances are (y  y 3 ) + (y 6  JVg), CVi Vi) + Cv? ~ J>io)>
(yz ~~ y&) + (y& ~~ y\\)> Draw a curve through the points,
and repeat below; and do the same thing again for the
second period of the wave.
To obtain the amplitude a", draw a tangent, and take
J of the distance to the x axis, = 25. To obtain the angle
, measure O?; Oq  of O3 = 150, hence = 210.
For the third harmonic. The successive distances for the
ordinates to be taken from the strip are, first (0 to 2) + (4 to 6)
+(8 to 10), then (1 to 3) + (5 to 7) + (9 to 11). The curve
almost coincides with the x axis, and as the distance to the
crests has to be divided by 6 to obtain the amplitude a'",
210* \
R25J
\
Fig. 265
it is evident that this third term of the harmonic series is
negligible. Hence the equation may be written:
y = 35 sin(* + 282) + 25 sin(2^ + 210).
Now draw the two harmonics to scale (fig. 265), recompose,
compare the result with the original graph, and thus check
the work.
WAVE MOTION
469
Boys are always keen to know what is behind such un
usual procedure. The explanation is really very simple.
Let (i), fig. 266, be the first harmonic (the fundamental),
(ii) the second harmonic, (iii) the third, and (iv) the fourth.
We can cut the whole wave in (ii) into two equal and similar
parts, and slide the righthand half along the axis and superpose
it on the lefthand half. We may cut (iii) into three equal and
Fig. 266
similar parts, slide the second and third parts along and
superpose them on the first. We may cut (iv) into four
equal and similar parts, and again superpose.
Now suppose we have a compound curve of unknown
composition. If it consisted of the first harmonic only, it
would be just a simple sine curve, like fig. (i).
If the second harmonic is present, fig. (ii) represents that
component. To test for its presence, cut the composite curve
to 2?r into two, slide along and superpose, add the corre
sponding ordinates of the two parts thus superposed (yo+JVe*
y l + y 7 , &c., algebraically, of course), take the average of
each of these sums by dividing by 2, and plot the curve.
That curve is the second harmonic together with any of its
multiples, if any of these are components; but the curve
470 CRAFTSMANSHIP IN MATHEMATICS
does not contain any other harmonic than these multiples;
i.e. the curve so obtained is,
y = a 2 s'm(2x + oc 2 ) + 4 sin(4# f a 4 ) f &c.
If the third harmonic is present, fig. (iii) represents the
component. To test for its presence, cut the composite curve
to 2?7 into three, slide along and superpose, add the corre
sponding ordinates of the 3 parts thus superposed, take the
average of each sum by dividing by 3, and plot the curve.
The curve is the third harmonic together with any of its
multiples, if any of these are components, but the curve
does not contain any other harmonic than those multiples;
i.e. the curve obtained is,
y = 3 sin(3# f oc 3 ) + a 6 sin(6# f a e ) f &c.
So with harmonics beyond the third. But these are
rarely required; they affect the result too slightly. The
proofs of these rules are very simple, and should be given.
Inasmuch as there is no advantage in giving for analysis
any composite curves containing harmonics beyond the
third, this graphic work need not be carried further. But
the boys ought now to return to the example represented by
figs. 263 and 264, and penetrate the mystery of the paper
strip: the additions from the strip are really the additions
of superposed ordinates resulting from cutting up the com
posite curve, sliding to the left, and superposing. The
reversal of the strip is readily seen to be a simple device for
converting subtraction into addition.
Teachers who think well of this method of Professor Runge
may refer to Zeitschrift fur Mathematik und Physik, Vol. 48,
44356.
Professor Nunn's Plan: the Principle
A much more important curve decomposition method
may be briefly considered. The fundamental principle
WAVE MOTION
471
underlying it is the obvious fact that the total area of either
a complete sine curve or of a complete cosine curve is zero,
since it is equally divided by the x axis. Professor Nunn's
exposition (Algebra, pp. 5213) is particularly illuminating,
though I have sometimes found Sixth Form boys, who had
Fig. 267
not had a good training in solid geometry, puzzled over the
geometrical figures. I append an outline of the exposition,
together with a few new " solid " figures.
On one side of a line AB of length /, draw the semi
sine curve y = a sin*,
choosing any value for
the amplitude HK (=
a). On the other side
draw similarly the curve A
A
y = sin*, with ampli
tude KL (= unity). Cut
the figure out and fold
it about AB until the
planes of the two curves Fig. 2 68
are at right angles. Now
mould a solid, in clay, plasticine, soap, or any similar soft
material, to fill up the space between the curves.
In practice, the best way to do this is first to mould a
rectangular prism / units long with crosssection KH X KL.
Then draw the curve a sin ^* on the face of the prism
FCDE (i), and the curve sin* on the top of the prism
MFEG (ii). Pare off horizontally round the curve as in (i),
r
472
CRAFTSMANSHIP IN MATHEMATICS
and vertically round the curve as in (ii). The result is (iii),
the solid we require, ALBH; the plan of the solid is the
77
figure bounded by AB and the curve sin*, and the elevation
in the figure bounded by AB and the curve a siny#.
It is important to note that any section of the solid by a
plane at right angles to AB is a rectangle (e.g. RSTV) whose
adjacent sides are a sin# and sin#, x being the distance
i if
of the section from A. Note that the two lengths may be
measured either on the flat surfaces behind and below or
on the curved surfaces in front and above. Unless the solid
is actually constructed, many boys will have difficulty in
seeing this.
The area of the section = a sin nx/l X sin nx/l
= a sm 2 Ttx/l
= 2(1 cos2nx/l)
i:7*
WAVE MOTION 473
that is, the area of any section of the solid is equal to the
algebraic difference between a constant area  and a variable
2
a
area cos 27rx/l.
2
For convenience, each of these areas may be looked upon
as rectangles, each of height a/2. Thus the base of the
former would be unity, and that of the latter
I
B
(H)
cos 2ncc/l
I '
Fig. 270
The two rectangles may be regarded as crosssections
of two new solids of length AB ( /) and of uniform height
a/2. Above are their plans. Note the neat, though obvious,
(i)
(ii)
device for showing the width of the second. Fig. 271 shows
perspective sketches (for the sake of clearness, figs. 270 and
271 are drawn very considerably out of proportion, compared
with figs. 268 and 269).
474 CRAFTSMANSHIP IN MATHEMATICS
Let a plane at right angles to AB cut the solids at PiP 2 >
corresponding to RSTV in fig. 269 (iii). Then the section
RSTV is equal to the difference between the sections P 1 and
P 2 ; and so with any other vertical section. At KL in fig. 269
(iii), the difference is between Q x and Q 2 , but since Q 2 is
negative, the difference is the arithmetical sum. This is as
might be expected, for the section on HL is the full section
of the original rectangle. In the case of section P 2 , the width
cos 27rx/l is positive; in the case of Q 2 , it is negative. Thus
the area of the section P 2 must be reckoned positive and that
of Q 2 negative.
It follows that the part of the solid above AB in fig. 270
(ii) must be reckoned positive, and that below AB negative.
Hence we must regard the total volume of the solid in fig.
271 (ii) as 0. But the volume of the solid in fig. 269 (iii) is
equal to the difference of the volumes of the two solids in
fig. 271 (i) and (ii). Hence the volume of the solid in fig. 269
(iii) is equal to the volume of the simple prism in fig. 271 (i).
The volume of the solid in fig. 269 (iii) is therefore al/2. This
result is always a surprise to the boys, and they are much
inclined to question it. They should be made to think about
it carefully and to search for the fallacy they suspect. It
will pay to make the boys work out one or two particular
cases. Let them bear in mind that the volume of the rect
angular blocks in fig. 269 (see fig. 268) is / X a X 1 = al.
On one or two occasions I have known Sixth Form
boys cut out their models so carefully that, when checked
by weighing, the results have been surprisingly accurate.
To cut fig. 269 (iii) out of soap, and to weigh the model
against the parings, may afford a very convincing check.
WAVE MOTION
475
The Principle Applied
Consider the following figure, one complete element
(O to 2?r) of a composite wave. The problem is to determine
the amplitudes of the various component harmonics; that
done, the harmonics are easily drawn. Since the righthand
half of the curve is the " image " of the lefthand half, it is
sufficient to consider the lefthand half alone; call its length
/. We will assume that there are two components, viz. y = a v
simrx/l and y = a 2 smS^x/l, in other words that the given
curve is made up of the first and second harmonics. (We
know from the kind of symmetry that the third harmonic is
not a component (see p. 463).)
On the line IVTN' (= MN = /), draw the curve y =
sin irx/l inverted, i.e. a sine curve with amplitude unity; and
Fig. 272
make a model of the solid determined by the two curves
when the lines M'N' and MN are made to coincide and
the planes of the figures are at right angles. Note that any
section FGK at right angles to MN is rectangular, as in the
solid of fig. 269. The solid is not an easy one to model
accurately (fig. 273).
The volume of the composite solid is equal to the
476 CRAFTSMANSHIP IN MATHEMATICS
sum of the two solids determined by the curves,
(i) y = smnx/l and y = a^ smnx/l,
and (ii) y = sin roe// and y = a z s'm2nx/l.
But the latter of these volumes is easily proved equal to
(cf. fig. 271 (ii)), and the volume of the former is a x //2 (cf.
fig. 271 (i)). Hence
total volume of solid = aJ/2 (i)
Fig. 273
But the volume may also be determined directly, by
calculating the mean value of its crosssection. Consider, for
instance, the vertical section at G on MN where MG = 2//3
= x, so that TTX/I = 277/3 radians or 120. The section in a
rectangle whose sides FG, GK are closely analogous to the
sides RS, ST in fig. 269 (iii). Of these two sides, FG may
easily be determined by actual measurement from the curve,
while GK = sin!20 = \/ 3 / 2  The product gives the area
of the vertical section through FGK.
In this way we may find the area of any number of such
vertical sections. For convenience, divide MN into 12 equal
parts, Calculate the areas of the respective sections through
WAVE MOTION 477
the dividing points, and then by Simpson's rule* the volume
of the solid. Deduce from this the average cross section
A x by dividing by /.
Thus vol. = A!/.
But vol. = aJ/2 (by (i)).
" ll  A /
T~ Al/ *
or flj = 2Aj.
In a similar manner, by supposing a second solid to be
formed by combining the given half curve with the curve
y = sin2?rjc//, the value of a 2 may be determined. If the
given curve contained any other harmonic components,
their amplitudes might be determined in the same way.
The principle of the method is that any sine curve y =
sin rnx/l when combined with half the given composite curve
determines a solid whose volume (a r l/2) depends on the
amplitude a r of the component y = a r sin mx/l, and not at
all on the amplitude of any other component. In this way,
the successive sine components can be dealt with one by
one, and their amplitudes determined. The determination
of the amplitudes is, of course, the very essence of the problem.
The work of computing the average crosssections can be
divided up amongst the members of the class. Instruct them
to carry out the following operations, and to tabulate the
results:
(1) To determine the amplitude of the first harmonic.
(a) Divide up MN into twelve 15phase differences;
erect the ordinates and measure their lengths in millimetres.
In accordance with Simpson's rule, only half the height
of the first and last ordinates is required in the calculations,
but as, in this instance, these happen to be zero, the halving
makes no difference.
(/J) Calculate the successive values of sinn!5, n being the
number of the ordinates.
* " Add half the first and last areas and the whole of the intermediate areas,
and multiply the sum by the common interval."
478
CRAFTSMANSHIP IN MATHEMATICS
(y) Multiply (a) by (/?) and so obtain the areas of the
successive sections of the solid (fig. 273).
i.
ii.
1st Harmonic.
2nd Harmonic.
Number
of
Ordinate.
Length of
Ordinate
in mm.
iii.
iv.
V.
vi.
vii.
viii.
Angle.
Sine.
Area
ii or iv.
Angle.
Sine.
Area
ii or iv.
+o
1
130
15
26
338
30
5
+ 650
2
230
30
5
1150
60
87
+ 1993
3
295
45
71
2090
90
10
42950
4
315
60
87
2740
120
87
42740
5
295
75
97
2860
150
5
41475
6
240
90
10
2400
180
7
175
105
97
1690
210
5
875
8
103
120
87
892
240
87
892
9
50
135
71
354
270
10
500
10
20
150
5
100
300
87
174
11
10
165
26
26
330
5
50
12
180
360
Total =
Average area A x =
14640
Total =
7316
1220
Av'ge = A 2 =
6097
Amplitude of the 1st harmonic =
fll = 2Aj = (122 mm. X 2)
= 244 mm.
= 244 cm.
(2) To determine the amplitude of the second harmonic.
Corresponding to the half curve MN will be a complete
sine curve of the second harmonic. Hence the angles will
now be #30, and the sines from 180 to 360 will be negative
The ordinate lengths will be the same as before.
Amplitude = a a =
= (6097 mm. X 2)
= 122 cm.
WAVE MOTION 479
Hence the original curve is,
y = 244 sinnx/l+ 122 sin2nx/l.
The periods being known, and the amplitudes having been
found, the angles follow at once.
Let the boys realize fully that the essence of the problem
is the discovery of the amplitudes of the component harmonics.
315
274
By improvising the solids and devising two different schemes
for determining their volumes, we obtain two different formulae
each involving a in terms of A. It is true that A appears as
an area, but, by taking one of the dimensions of the area as
unity, A becomes a linear value, and of course we begin by
giving the solid a base consisting of a sine curve of unit
amplitude.
The subject can be followed up by integration. The boys
are now ready for it, for they have learnt what the subject
is really about.
Books to consult:
1. The Teaching of Algebra, Nunn.
2. Manual of Practical Mathematics, Castle.
3. Any modern standard work on Sound.
480 CRAFTSMANSHIP IN MATHEMATICS,
CHAPTER XXXII
Mechanics
The Teacher of Mechanics
The most successful teachers of mechanics whom I have
known are those who have had a serious training in a me
chanical laboratory; who know something of engineering, and
are familiar with modern mechanism; who are competent
mathematicians; and who have mastered Mach's Mechanics,
especially Chapters I and II.* Mach's book is universally
recognized as the book for all teachers of mechanics. It
deals with the development of the fundamental principles
of the subject, traces them to their origin, and deals with
them historically and critically. The treatment is masterly.
The book might with advantage be supplemented by Stallo's
Concepts of Modern Physics (now out of date from some
points of view), Karl Pearson's Grammar of Science, and
Clifford's Common Sense of the Exact Sciences and Lectures
and Essays (still firstrate, though written 50 years ago).
It is of great advantage to a teacher of mechanics to be
familiar with the subject historically. The main ideas of the
subject have almost always emerged from the investigation of
very simple mechanical processes, and an analysis of the
history of the discussions concerning these is the most
effective method of getting down to bedrock.
Who were the great investigators? The scientific treat
ment of statics was initiated by Archimedes (287212 B.C.),
who is truly the father of that branch of mechanics. The
work he did was amazing, but there was then a halt for 1700
or 1800 years, when we come to Leonardo, Galileo, Stevinus,
and Huygens; to Torricelli and Pascal; and to Guericke
and Boyle. For dynamics, we go first to its founder Galileo
* Hertz also wrote a Mechanics of the same masterly kind, but there is no English
translation, so far as I know.
MECHANICS 481
(falling bodies, and motion of projectiles), then to Huygens
(the pendulum, centripetal acceleration, magnitude of acceler
ation due to gravity), and then to Newton (gravitation, laws
of motion). The great principles established by Newton
have been universally accepted almost down to the present
time, and, so far as ordinary school work is concerned, will
continue to be used at least during the present generation.
A boy is always impressed by Newton's argument that
since the attraction of gravity is observed to prevail not
only on the surface of the earth but also on high mountains
and in deep mines, the question naturally arises whether it
must not also operate at greater heights and depths, whether
even the moon must not be subject to it. And the boy is
still more impressed by the story of the success of Newton's
subsequent investigation.
Newton's four rules for the conduct of scientific investi
gation (regulce philosophandi) are the key to the whole of his
work, and should be borne in mind by his readers.
The First Stage in the Teaching of Mechanics
How do successful teachers begin mechanics with boys of
about 12 or 13? They usually begin by drawing upon the
boys' stock of knowledge of mechanism.* Most boys know
something of mechanism, some will have had enough curiosity
to discover a great deal, and a few will probably have had
experience of taking to pieces machines of some sort and of
putting them together again. This stock of knowledge may be
sorted out, and the topics classified and made the subjects
of a series of lessons. By means of an informal lesson on some
piece of mechanism, an important principle may often be
worked out, at least in a rough way.
I have known a teacher give his first lesson on mechanics
in the school workshop, utilizing the powerdriven lathe and
the drillingmachine; another first lesson in the school play
ground, an ordinary bicycle being taken to pieces. I have
See Chapter VIII, Science Teaching.
(291) 32
482 CRAFTSMANSHIP IN MATHEMATICS
seen a model steamengine used for the same purpose,
and I have known beginners taken to a local farm to watch
agricultural machinery at work. In all these instances the
boys learnt that their new subject seemed to have a very
close relation with practical life. They were not made to
look upon it as another branch of mathematics, and a rather
difficult branch at that.
Let the early lessons be lessons to establish very simple
principles. Never mind refinements and very accurate
measurements. Do not bother about small details, and avoid
all complications. Let the boy get the idea, and get it clearly.
Very simple arithmetical verifications are quite enough at
this stage. The boy's curiosity is at first qualitative; let
that be whetted first, and then turned into a quantitative
direction gradually. Encourage the boy to find out things for
himself, and do not tell him more than is really necessary.
Encourage him to ask questions, but as often as possible
answer these by asking other questions which will put him
on a new line of inquiry. Let him accumulate knowledge
of machines and machine processes. Give him some scales
and weights, and a steelyard, and tell him just enough to
enable him to discover the principle of moments, but do
not talk at first about either " principle " or " moments ".
It is good enough if at this stage he suggests that
long arm X little weight  short arm X big weight.
He has the idea, and the idea is expressed in such a form that
it sticks. Give him a model wheel and axle, give him a hint
that it is really the lever and the leverlaw over again, and
make him show this clearly. Give him some pulleys and let
him discover, with the help of one or two leading questions,
how a small weight may be made to pull up a big weight,
and let him work out the same law once more, but now in the
form that what is gained in power is lost in speed. Give him
a triangular block and an endless chain, let him repeat Ste
vinus' experiment, and so discover the secret of the inclined
plane. Let him use a jack to raise your motorcar (and inci
MECHANICS 483
dentally learn something about " work "); now tell him some
thing about the pitch of the screw, something about Whit
worth's device for measuring very small increases in length,
something about the manufacture of a Rowlands grating.
Encourage him to give explanations of mechanical happenings
in everyday life, and use his suggestions as pegs on which to
hang something new.
A term of this kind of work pays. The boy is accumulating
knowledge of the right sort, and when the subject is taken up
more formally and with a more logical sequence, rapid progress
may be made. Once he has been taught to read elementary
mechanism, it is easy enough to teach him its grammar.
Surely this is the right sequence. Mechanism must come
before mechanics. The mathematics of the subject is a super
structure, to be built upon a foundation of clear ideas.
Of course, if the preliminary work of the preparatory school
or department has been properly done, the way is paved for
an earlier treatment of a more formal kind.
The Second Stage
The second stage should consist of work of a more syste
matic character, but still work essentially practical, though
arranged on a logical string. Ideas will now be classified, and
mathematical relations gradually introduced. But the physical
thing and the physical action must still remain in the front of
the boy's mind. The mathematics will take care of itself.
Let the teaching be inductive as far as possible. Obtain
all .necessary facts from experiments, and do not use experi
ments merely for verifying a principle enunciated dogmatically.
The basic principles to be taught are really very few, and
a boy who knows these thoroughly well can work most ordinary
problems on them. Mechanics is, after all, largely a matter of
common sense. The laws of equilibrium, together with the
ratio of stress to strain, covers almost the whole range of
statical problems, including those of hydrostatics; while New
ton's Laws of Motion covers practically everything else. But
484 CRAFTSMANSHIP IN MATHEMATICS
of course these are basic principles. If they are known,
known, derived principles are learnt easily enough; if they are
only vaguely known, derived principles are never really
mastered.
Statics or dynamics * first? Teachers do not agree. There
is much to be said for beginning with dynamics, first using
the ballistic balance for studying colliding bodies, and the
momentum lost by one and gained by another; it is then an
easy step to pass on to the idea of force. But a boy who is
led to think of a force as something analogous to muscular
effort will always be in trouble, and in any case he is likely
to form a very vague idea of acceleration. Of course, uniform
acceleration is anything but common in practical life: we
nearly always refer either to falling bodies or to a train
moving out from a station. And it is this difficulty that
makes many teachers take up statics first. Although, at the
outset, a boy's working idea of force is necessarily crude,
a spring balance, for simple quantitative experiments, helps
to put the boy on the right track, and there is much to be
said for allowing him to assume, to begin with, that weight
is the fundamental thing to be associated with force. At
an early stage he may verify, to his own satisfaction, the
principles of the parallelogram and triangle of forces, but he
must be warned that he has not yet " proved " these principles
and cannot yet do so. But since the parallelogram of forces is
such a useful working principle, it would be foolish not to
allow the boy to use it before he can prove it formally. At
this stage formal proofs are difficult, and it is simply dis
honest to encourage a boy to reproduce a page of bookwork
giving a proof of something quite beyond his comprehension,
though this was common enough thirty or forty years ago.
Do not employ graphic statics at too early a stage, or the
real point at issue may be obscured.
Now as to dynamics. What is the best approach? We
have already referred to the ballistic balance. Should At
wood's machine be used? It may be used, perhaps, for
* The terms kinetics and kinematics are falling into disuse.
MECHANICS 485
illustrating the laws of motion, but not as a practical method
of finding g.
Atwood's machine has been superseded by Mr. Fletcher's
trolley,* by means of which practically the whole of the prin
ciples of dynamics may be satisfactorily demonstrated. It
lends itself to many experiments, all of which provide a
spacetime curve ready made, and, from that, speedtime
and accelerationtime curves may be plotted. In a paper
read at the York meeting of the British Association, Mr. C. E.
Ashford gave details of a large number of trolley experiments
as performed at Dartmouth, a school where the teaching
of mechanics is well known to be of a high order. Reference
should be made to Mr. Fletcher's own article in the School
World for May, 1904. In it he shows how boys may be
given sound ideas of the physical meaning of the terms,
moment of inertia, angular momentum, moment of momen
tum, and therefore of moment of rate of change of momentum
and moment of force. Useful teaching hints may also be found
in Mr. S. H. Wells's Practical Mechanics and Mr. W. D.
Eggar's Mechanics.
Once the foundations of mechanics have been well and
truly laid the superstructure may be erected according to
traditional methods. To leave the subject just as developed
in the laboratory would be to leave it unfinished. But the
superstructure may now be built properly. When necessary
formulae have been evolved from experiment, the physical
things behind the formulae have to the boy a reality of mean
ing which the older " methods of applied mathematics "
teaching could not possibly give him.
If principles are not understood, proofs have no meaning.
Throughout the whole of a mechanics course every oppor
tunity should be taken to excite the boys' interest in new
mechanical inventions. It helps the more academic work
*The friction of the trolley may be eliminated either by tilting the plane to the
necessary angle, or by attaching a weight that will just maintain uniform motion.
The friction of the pulley over which the thread passes cannot be compensated, and
it is therefore necessary to use a good pulley.
486 CRAFTSMANSHIP IN MATHEMATICS
enormously, and makes the boys feel that the subject is really
worth taking trouble over. Examples occur on every side
variable speed gears, transmission gears, taximeters, boat
lowering gear, automatic railway signalling, automatic tele
phones, the selfstarter in a motorcar, the kickstarter in a
motorcycle, and so on. Some mechanical devices depend, in
their turn, on electricity, and their place of introduction into
a teaching course would be determined accordingly. Complex
mechanisms like the airplane, the submarine, the paravane,
should not be wholly forgotten. Boys can read up such things
for themselves, and perhaps prepare and read papers on them
to the school science society.
Hydrostatics
The mechanics of fluids is an exceedingly difficult subject
to teach effectively. Even a Sixth Form boy is sometimes held
up by questions on the barometer or on Dulong and Petit's
equilibrating columns. The work of Archimedes and Pascal
for liquids and of Boyle for gases cannot be too well done.
Above all, the Utube must receive careful attention, and
especially the surface level above which pressures are compared.
Do not buy Hare's apparatus from an instrumentmaker's.
The standard pattern is always made with two straight tubes,
of the same bore, fixed vertically. Let the boys make a variety
of forms of this apparatus for themselves, and work out the
vertical height law from data as varied as possible. Approach
the whole subject of hydrostatics from the point of view of
familiar phenomena, e.g. measure the water pressure from a
tap in the basement and again from a tap in the top story of
the school, and see if there is any sort of relation between the
difference of these pressures and the height of the school.
Do not try to establish a principle formally until the phenomenon
under investigation is clearly understood as a physical happening.
Let boys know really what they are going to measure before
they begin to measure.*
The preceding paragraphs are taken from Science Teaching (pp. 1218).
MECHANICS 487
The Johannesburg British Association Meeting
At the Johannesburg meeting of the British Association,
an animated discussion took place on the general question
of the teaching of mechanics. It followed on a paper read by
Professor Perry. We append a few suggestive extracts.
Professor Perry. " The very mathematical man often
does not know anything of mechanics; it is the subject of
applied mathematics that he has studied and that he cares
for.
" The two elementary principles of statics, (1) if forces
are in equilibrium, their vector sum is zero, and (2) the sum
of their moments about any axis whatsoever is zero, ought
to be so clear to a pupil that it is practically impossible for
him to forget them. They ought to be as much a part of his
mental machinery as the power to walk is part of his physical
function.
" I lay no stress upon mere abstract proofs of propositions
in mechanics. When understanding is affected there is no
difficulty about the proofs. It is quite usual to find men who
can prove everything, without having any comprehension of
what they have proved/'
Mr. W. H. Macaulay. " I agree with the taking of statics
before dynamics. I also agree that graphical statics is a subject
full of dodges, though very good to learn if you want to use
them every day."
Professor Boys. " I absolutely agree as to the desirability
of dealing with fundamental principles, and of not worrying
about innumerable details. ... A friend of mine heard
Lord Kelvin say in one of his lectures, ' And now we come
to the principle of the lever. You will understand that levers
are divided into three orders, levers of the first order, of the
second, and of the third but which of them is which I
cannot for the life of me tell you/ Textbooks were at one
time filled up with futile and unnecessary kinds of dis
crimination which had nothing whatever to do with the sub
ject."
488 CRAFTSMANSHIP IN MATHEMATICS
Professor Bryan. " The idea of mechanics which appeals
most readily to a young boy is that it has something to do
with machines, and that machines have something to do with
turning out useful work. There is no better way of stimulating
interest in the subject than showing the beginner that when
you have got your machine for changing one kind of work
into another, you are no better off than when you started. "
Professor Hicks. " My own experience is in approaching
mechanics from a kinetic point of view. First let the boys
find out by experiment that momentum remains constant.
Of course the first thing depends on what mass is; then we
must proceed to show that when two bodies collide with equal
velocities they come to rest. By making experiments of
velocities of colliding bodies, boys get to realize that momentum
remains unalterable. Given two colliding bodies in a straight
line, the momentum lost by one is gained by the other.
By getting a large number of experiments, pupils come to
a realized knowledge of that."
Sir David Gill. " I remember Clerk Maxwell illustrating
the misuse of definitions by a funny story. He said he went
into his room one day, and there was a white cat which
jumped out of the window. He and his friends ran to the
window to see what had become of the cat, and the animal
had disappeared, no one being able to solve the mystery.
At last he solved the problem. He said it must be this. The
white cat jumped out of the window, fell a certain distance
with a certain velocity, and collided with an ascending black
cat. There were therefore two equal and opposite cats
meeting with equal and opposite velocities, the result being
no cat. Without a proper understanding of definitions of
these things, one might arrive at such an absurdity as this
story illustrates."
Professor Forsyth. " The first stage in teaching mechanics
is not the stage in which pupils have to prove, or attempt to
prove, or can be expected to prove, anything. That belongs
to a later stage. The first thing to do is accustom the pupils
to the ordinary relations of bodies and of their properties."
MECHANICS 489
Mr. W. D. Eggar. " I should like to see a pennyin
t heslot automatic weighing machine in every passenger lift,
so that the fundamental experiment of showing a connexion
between force and acceleration could be within the reach of
everybody."
Professor Minchin. " I hope to see the term * centrifugal
force ' utterly banished."
Mr. C. Godfrey. " Statics is a fairly easy matter if one
begins with experiment. Nor need experiment cease after
the first stage; any school should be able to get hold of some
bit of machinery with plenty of friction in it, say a screw
jack, and investigate efficiency. Plotting ' load ' against
* effort ' leads to very striking results.
" There is the question of mass and weight. In vain one
resorts to the centre of the earth; it is all too hypothetical.
I remember as a boy being puzzled to understand how the
weight of a train (acting vertically) could have anything to
do with its acceleration under a pull (horizontal) from the
engine.
" We might give a touch of reality to the kinetics course
by brake horsepower determinations. It should be possible
to rig up for a few shillings a brakedrum on a motor (electric
or water); even a motorcycle on a stand or a footlathe
might serve the purpose.
" Engineers talk in a very confusing way about centrifugal
force. When a particle moves in a circle uniformly, the force
on the particle is centripetal and the force on the constraints
is centrifugal. But the popular use of language and the
popular belief is that there is an outward force on the particle."
" Applied " Mathematics
The old school of " pure " mathematicians very cleverly
picked out from the whole subject of mechanics and engineer
ing such problems as lent themselves to algebraic and
geometrical treatment, and left the residue, rather disdain
fully labelled " applied mechanics ", to be dealt with by
490 CRAFTSMANSHIP IN MATHEMATICS
teachers of lower degree. Note the term " applied ". The
real mechanics was the mechanics that could be done from
an easy chair, and was a mathematicians' job. The building
of the Assouan dam and of the Forth Bridge were trivial
things which any " ordinary engineer " could take in hand,
trivial things that had no relation whatever to " pure "
thought. This temper survived even until the present century.
When the two Wrights were risking their lives by experiment
ing with the first airplane, a wellknown mathematician wrote
to the press protesting against such folly, inasmuch as mathe
maticians had not yet worked out the mathematical principles
of flight!
The mathematician's proper share of such work is to
begin where the inventor or the engineer leaves off; it is
not his business to invent paper airplanes, but to learn
from the real thing the principles of flight and to see if these
rest on secure mathematical foundations; if they do not,
he may be able to offer fruitful suggestions. Of course if
the mathematician happens to have been trained as an
engineer, that is a different matter. Unless the teacher of
mechanics knows something of actual engineering, his me
chanics is likely to have but a remote connexion with actual
mechanism. There are still teachers of mechanics who have
had neither workshop nor laboratory experience, and naturally
they tend to shirk those parts of the subject that do not
come within the four corners of algebra and geometry. It
is not an uncommon thing for a course of lessons on ele
mentary statics to include not a single word about, for instance,
the equilibrium and stability of walls, the effect of buttresses,
the thrust along rafters, or about rooftrusses or cranes.
Friction may be the subject of a lesson with no mention
whatever of lubricants. Energy may be the subject of others,
and yet no reference be made to energy storage in, for example,
accumulators and flywheels. The transmission of motion
and power is rarely touched upon seriously in a course of
mechanics lessons. And yet all such things as are thus ignored
are just those things that have already been included, in
MECHANICS 491
some measure, within the four corners of the boys' daily
experience. Subjects like tension and compression, shearing
and torsion, beams, girders, and frameworks, are passed
over hurriedly as of little importance. Why is elementary
hydrostatics so often given such short shrift? Why is it not
followed up by the subject which really matters, viz. ele
mentary hydraulics the flow of water through orifices and
pipes, the pressure in a watermain, waterwheels, turbines,
the propulsion of ships and airplanes, and hydraulit machines?
As for capillarity and surface tension, which lend themselves
to all sorts of delightful experiments, they are too often an
affair of just blackboard and chalk. Do not put off that
interesting section of physics, " properties of matter " (the
twinsister of mechanics), until the Sixth Form. The mathe
matics of it in the Sixth, yes; but the necessafy laboratory
course can be taken in the Fourth and Fifth.
In short, the mathematics of mechanics is very serious
Sixth Form work. The practical work that must be done
before the mathematical work can profitably be attempted
may be done earlier.
We will give an extract from an elementary textbook
on Mechanics for Beginners, with a very well known name
on the titlepage. It is an introduction to Moment of
Inertia.
" Let the mass of every particle of a body be multiplied
into the square of its distance from an assigned straight line;
the sum of these products is called the moment of inertia of
the body about that straight line. The straight line is often
called an axis.
" The moment of inertia of any body about an assigned
axis is equal to the moment of inertia of the body about a
parallel axis through the centre of gravity of the body, increased
by the product of the mass of the body into the square of the
distance between the axes. Let m be the mass of one particle
of the body; let this particle be at A. Suppose a plane through
A, at right angles to the assigned axis, to meet the axis at
492 CRAFTSMANSHIP IN MATHEMATICS
O, and to meet the parallel axis through the centre of gravity
A at G. From A draw a straight line
AM, perpendicular to OG or to OG
produced. Let GM = x, where x is
a positive or negative quantity accord
ing as M is to the right or left of G.
"M By Euclid II, 12, 13, we have OA 2
= OG 2 + Ga 2 + 2OG.*; therefore,
m . OA 2 = m . OG 2 + m . GA 2 + 2OG . m . x.
A similar result holds good with respect to every particle of
the body. Hence we see that the moment of inertia with
respect to the assigned axis is composed of three parts,
namely, first the sum of such terms as w.OG 2 , and this will
be equal to the product of the mass of the body into OG 2 ;
secondly, the sum of such terms as m.Ga 2 , and this will be
the moment of inertia of the body about the axis through
G; and thirdly the sum of such terms as 2OG .m.x y which is
zero. Hence the moment of inertia about the assigned axis
has the value stated in the proposition. "
Be it remembered that this book is a book for "beginners".
I remember a Fourth Form once being given ten minutes
to read up the subjectmatter just quoted. Then came
questions. Said one boy, " I thought inertia meant lazi
ness. " " So it does, a sort of laziness."" Then does
' moment of inertia ' mean a moment of laziness?" Said
another boy, " How are we to find the mass of one particle?
Do we crush the thing up in a mortar, and weigh one of the
particles? or do we weigh the thing first, then crush it up,
count up the particles, and divide the weight by the number?"
The teacher replied, " Don't be silly; moment of inertia is
not real; it is only theory "!
Who could blame the boys for asking such questions?
How could they have obtained the faintest insight into the
nature of the subject under discussion?
Forty or fifty years ago, Todhunter's Analytical Statics
was a standard work, used by mathematical students at the
MECHANICS 493
University. There are cases on record of men who obtained
Firsts in mathematics but who in the subject mentioned had
read no other book at all, had never handled a piece of appa
ratus in their lives. Fortunately that age has passed away.
Mr. Fletcher's trolley, which is now in general use for
teaching dynamics, is not always made so serviceable as it
might be. (Readers should refer again to Mr. Fletcher's
own comprehensive article in the School World for 1904.)
In Perry's Teaching of Elementary Mechanics, already referred
to, Mr. Ashford, formerly Head of the Royal Naval College,
Dartmouth, gives some exceedingly useful hints on the
further use of the trolley.
The early teaching of mechanics must be given an ex
perimental basis. Mathematicians unacquainted with the
mechanical laboratory should let the subject alone. It is
better not taught at all than to be taught as mere algebra and
geometry. Only if basic principles are established experi
mentally can the subsequent mathematical work be given
a reality and a rigour that command respect.
44 The Teaching of Mechanics in Schools "
A report on " The Teaching of Mechanics in Schools ",
specially prepared for the Mathematical Association, was
issued in 1930. The responsible subcommittee was appointed
in 1927 by the General Teaching Committee of the Association.
The subcommittee included such wellknown teachers as
Mr. C. O. Tuckey, Mr. W. C. Fletcher, Mr. W. J. Dobbs,
Mr. C. J. A. Trimble, and Mr. A. Robson, and the Report
will therefore carry great weight amongst all teachers of
mathematics. Every page reveals the hand of the practical
teacher. No teacher of mathematics should fail to give it
his serious attention. We quote a few short paragraphs in
order that the reader may gather some notion of the general
tenor of the Report.
" There is perhaps no branch of mathematical instruction
for which a pupil comes prepared with a larger body of
494 CRAFTSMANSHIP IN MATHEMATICS
intuitional knowledge than he does for mechanics. The
suggestions made in this report are based on the view that
this body of knowledge should form the foundation of the
teaching, and that the aim of the teaching should be largely
concerned with a development of a taste for such accurate
thought and consideration of mechanical facts as will make
them more intelligible, increasing the interest which attaches to
the mechanical behaviour of things, and leading to that insight
which brings this behaviour more completely under control/'
" Just as geometry has its roots in familiar phenomena of
daily life, so has mechanics. The basic principles of both
sciences can be gathered, at least crudely, from ordinary
observation this is the process we knew as abstraction. "
" When we have carried the process some little way it
becomes necessary, or at least economical, to arrange things
so as to provide a more exact answer to a definite question
than can be obtained from observation of unarranged or
uncontrolled phenomena. So we get two processes, fading
into one another no doubt in marginal cases, but in
general easily distinguishable, viz. reflection on ordinary
experience, and deliberately arranged experiment. In the
former it may be noted and it is perhaps an essential part
of the distinction experience comes before thought; we
may or may not observe and reflect upon it and we may or
may not make scientific use of it. In the latter, viz. experi
ment, as it has to be deliberately arranged, thought comes
first we must frame a question before we can arrange the
experiment which is to give the answer. "
" While there is room for difference of opinion and
practice as to the place of experiment in the school treatment
of mechanics, there is no question that observation and
reflection on ordinary experience are essential for any proper
grasp of the subject. The widespread neglect of this obvious
truth is responsible for much lack of success in the teaching
of the subject."
" In mechanics the crude facts lie open to direct observa
tion, and the role of experiment is limited to rendering more
MECHANICS 495
precise an answer which, in the rough, can be given without
experiment."
" The function of experience is to provide a basis of reality
for the abstract science of the textbook and the schoolmaster,
and the paramount duty of the latter is to make his pupils
conscious of their own experience, to get them to reflect
upon it, to coordinate their existing store and to open their
eyes to observe more closely and to see the significance and
interest of much that the unobservant mind ignores. Training
of this sort is essential if the subject is to have its real
value. ... In each fresh section of the work, the first thing
to do is to collect and clear up existing experience bearing
on the matter in hand."
The " Contents " of the Report are as follows:
1. Position in the Curriculum.
2. General Aims.
3. Experience and Experiment.
4. Order of Treatment.
5. The beginning of Statics.
6. The beginning of Dynamics.
7 Miscellaneous Topics:
(i) Earlier teaching of Mechanics; (ii) Experiments; (iii)
Initial difficulty of Statics; (iv) Kinematics; (v) Units and
Dimensions; (vt) Horse Power; (vii) Formation of the
Equations of Motion; (viii) Jointed Frames; (ix) Friction;
(x) Torque, Couples; (xi) Geometrical and Algebraic
Methods; (xii) Impact and the Lew of Momentum
and Energy; (xiii) Rotatory Motion; (xiv) Limitations
of School Dynamics; (xv) Miscellaneous.
8. To examiners.
9. Appendices:
(i) Wheeled Vehicles; (ii) Momentum Diagram.
Newtonian Mechanics superseded
It is commonly said that Einstein has dethroned Newton,
and this in a sense is true, inasmuch as Newton's laws
have been superseded; but Einstein has always regarded
Newton as his master. Improved instruments have led to
496 CRAFTSMANSHIP IN MATHEMATICS
the discovery of facts unknown to Newton, and Newton's
laws have had to be amended in order that the new facts
may be included, and this has been really Einstein's work.
At the end of last century, physical science recognized
three indisputable universal laws: (1) conservation of matter;
(2) conservation of mass; (3) conservation of energy; and
on the strength of these laws physical science became almost
aggressively dogmatic. They should, of course, have been
regarded merely as working hypotheses. Since 1905, it has
been recognized that energy of every conceivable kind has
mass of its own. Mass is the aggregate of restmass and
energymass. Mass is seen to be conserved only because
matter and energy are conserved separately.
Then, again, as to the question of fixed axes. The trouble
that some of us had when learning mechanics in the days
of our youth arose (as we now see) from the assumption that
axes were fixed in space. It is impossible not to feel that
such able men as Kelvin, Tait, and Routh were not suspicious
that the theory was in some way incomplete, but they seem
to have acquiesced in giving to Newton's laws of motion
a universality and finality which we now know the laws
did not really possess.
Listen to Clerk Maxwell (as a mathematician probably
second only to Newton), in his lighter moments:
"RIGID BODY (Sings)
" Gin a body meet a body
Flyin' through the air,
Gin a body hit a body
Will it fly? and where?
Ilka impact has its measure,
Ne'er a ane hae I,
Yet a* the lads they measure me
Or, at least, they try.
" Gin a body meet a body
Altogether free,
How they travel afterwards
We do not always see.
MECHANICS 497
Ilka problem has its method
By analytics high;
For me, I ken na ane o* them,
But what the waur am I?"
How are the tremendously farreaching twentieth century
changes to affect our teaching? Probably not at all except
in the Sixth Form, for another twenty years to come. Of
course the changes are very slight, too slight to affect appreci
ably the actual practice of mechanics. But the theory of
mechanics is another story altogether.
Books to consult:
1. Mechanics, J. Cox.
2. Introduction to the Principles of Mechanics, J. F. S. Ross.
3. Theoretical Mechanics, J. H. Jeans.
4. Mechanics of Fluids, E. H. Barton.
5. Treatise on Hydrostatics, G. W. Minchin.
Routh, and Lamb, should still be on every teacher's shelf. Elemen
tary books like Ashford, Eggar, and Fawdry, are full of useful teaching
hints. The book for every teacher to master is Science of Mechanics
(Mach).
CHAPTER XXXIII
Astronomy*
Mathematics or Physics?
If astronomy is included in the school physics course,
the necessary mathematical work will be mainly supplementary.
If the subject has to be included wholly in the mathematical
course, it is not likely to have any great value. Mathematical
astronomy which is not based upon personal observations
*This chapter should be read in conjunction with Chapter XXVI of Science
Teaching.
(E291) 33
498 CRAFTSMANSHIP IN MATHEMATICS
of any kind, with the telescope at least, if not with the
spectroscope, is not likely to have much reality.
Elementary Work
A certain amount of introductory astronomy will neces
sarily be included in a school geography course. For
instance:
1. The earth as a globe travelling round the sun and
spinning all the time on its own axis inclined 661 to the
plane of the ecliptic, i.e. the plane of its path round the sun.
2. The consequences of these movements: day and night,
the seasons.
3. The moon as a globe spinning on its own axis once a
month, and travelling round the earth once a month, in a plane
slightly inclined to the plane of the ecliptic. Phases of the
moon.
4. Eclipses: comparative rarity of the phenomenon the
result of the inclination of the orbits of the earth and moon.
5. Fixing positions on the earth's surface. Latitude and
longitude. Elementary notions of map projection.
Older pupils who have done a fair amount of geometry,
especially geometry of the sphere, have no difficulty in under
standing these things from descriptions and diagrams. But
younger pupils require more help, otherwise they cannot visu
alize the phenomena, they remain puzzled, and their written
answers to questions are seldom satisfactory.
If an orrery is available, there is little difficulty, but more
often than not the teacher has to manage with improvised
models, perhaps a mounted globe to represent the earth, and
painted wooden balls to represent the sun and moon. Per
sonally I prefer to use a large porcelain globe (the kind used
with the oldfashioned paraffin lamps) to represent the sun,
the globe being fixed in position a foot or so above the centre
of the table, and illuminated from the inside by the most
powerful electric light available, the room being otherwise in
ASTRONOMY 499
darkness. This makes an admirable sun, and gives a sharply
defined shadow. The earth may be represented by a small
wooden ball painted white, with a knittingneedle thrust
through its centre to represent the axis, and with black circles
to represent the equator and the 23^ and 66^ parallels, the
ball being mounted so that its centre is the same height
above the table as is the centre of the sun, and the axis being
inclined at 66. About onehalf the " earth " is now brilliantly
illuminated, and the other half is in shade. If the earth is
moved round in its orbit, the successive positions of its axis
maintaining a constant parallelism, the meaning of (i) day
and night and their varying length in different parts of the
world, and (ii) the seasons, may be made clear in a few
sentences. If more serious work is to be done later, it is
particularly necessary that the plane of the ecliptic should
be clearly visualized, and this is easily done if the sun and the
earth are supposed to be half immersed in water, the surface
of the water representing the plane of the ecliptic. Make
the pupils see clearly that half the earth's equator is always
above, and the other half always below, this plane.
The phases of the moon are best taught by ignoring the
model of the earth for the time being and considering models
of the sun and moon alone. Let the laboratory sun illuminate
a painted ball, to represent the moon; let the pupils move
round this ball, from a position where they see the non
illuminated half to the position where they see the fully
illuminated half. One " phase " after another comes into
view, and further teaching is unnecessary. Now put the
"earth" in position, and show how the earth may get between
the sun and the moon, and prevent the sun from shining
on the moon; and how the moon may get between the earth
and the sun, and prevent our seeing the sun. And thus we
come to eclipses.
The first essential in teaching eclipses is to make pupils
realize that a cone of shadow is a thing of three dimensions.
Let the school sun cast the shadow of the much smaller
school earth. The whole classroom remains brilliantly lighted
500 CRAFTSMANSHIP IN MATHEMATICS
save for a cone of darkness on the far side of the earth (we
ignore all other objects in the room), and the shape and
size of this cone is easily demonstrated by holding a screen
at varying distances behind the earth. With a second ball
to represent the moon, correct notions of total, annular, and
partial eclipses may be readily given. It is quite easy to
show why eclipses are comparatively rare phenomena by
making the moon move round in an orbit inclined to the
earth 's orbit.
More Advanced Work
A Sixth Form ought to carry the subject very much
farther than the elementary aspects of it commonly included
in a geography course, but the business of the mathematical
teacher is not to give astronomy lectures in the wider sense
but to teach boys to solve those problems which are suggested
by the results of actual observation; for instance, the problem
of fixing the positions of the stars by means of their co
ordinates, the related question of the diurnal revolution of
the heavens, the daily movements of the sun and moon,
the calculation of times of rising and setting, nautical problems
of determining latitude and longitude, dialling problems.
Facts must not be confused with hypotheses. Thus the
earth's daily rotation on its axis and its annual revolution
round the sun are mere hypotheses, invented to account for
facts of observation. The mathematical teacher is concerned
with the face value of the facts observed. According to that
face value, the stars move round the sky daily, and the sun
and moon move amongst them. Any attempt to provide
a theory of stellar movements must be preceded by an exact
determination of the facts as they appear.
Quite low down the school the boys ought to have been
made familiar with the globe (a blackboard surface is very
useful) and a cardboard horizon fitting over it. And in the
very early stages of geometry they will have been introduced
to the theodolite, and will have been taught to measure
altitudes and azimuths (though perhaps the term azimuth
ASTRONOMY 501
has not been used). The theodolite may have been made in
the school workshop, and a mere cardboard tube used instead
of a telescope. But higher up the school an instrument
designed for fairly accurate measurements should be used,
and nowadays a good one may be purchased for a few pounds.
Even Fourth Form boys can be taught to measure the azimuth
and altitude of a given star as it appears to an observer at
a given moment. It is easy and interesting work and they
like it, though some of them seem to need repeated help
with the setting up and initial adjustment of the instrument.
I have known boys of 9 or 10 readily pick out the better
known constellations, and such stars as the Pole Star, Vega,
Capella, Sirius, and the Pleiades. This kind of observation work
ought to be included in every Nature Study course. It creates
an early interest that becomes permanent, and such basic
facts are very useful for future mathematical work.
A school lucky enough to have a small observatory of
its own will have an altazimuth (a theodolite is virtually
a portable altazimuth), so that azimuths and altitudes (or
zenith distances) may readily be found. An equatorial may
also be available. If not, the altazimuth should be of such
a kind that its telescope can be mounted equatorially when
required. Then the boys can take Declinations and Right
Ascensions, and become familiar with the celestial equator
as well as with the celestial pole, and they will then soon
look upon the rotating northern celestial hemisphere as an
old familiar friend. Once they feel this familiarity, the
making of reasonably accurate observations is child's play
and the mathematics involved is not difficult. The sidereal
clock and sidereal time are also easily mastered.
The solution of the common problem of determining the
altitude and azimuth of a star when the hour angle and
declination are given (or vice versa) is an easy case of the
solution of a spherical triangle, and should be familiar.
The sundial cannot profitably be taken up until the
Sixth, and not even then unless the boys have been well
grounded in the geometry of the sphere and its circles. The
502 CRAFTSMANSHIP IN MATHEMATICS
geometrical method of graduating the dial (to be fixed either
horizontally or on a south wall) is simple enough if the
elementary geometry of the sphere has been mastered. The
boys must be able to see that the key to the whole thing lies in
the fact that the edge of the gnomon is parallel to the earth's
axis and is therefore pointing in the direction of the celestial
pole. If about this they are vague, the whole thing is vague.
There is no better way of introducing the young observer
to the knowledge of the law of the sun's rotation than by
leading him to see that, if a dial be so placed that the style
(the edge of the gnomon) is parallel to the axis of the rotating
celestial hemisphere, the shadow of the style will at all seasons
of the year move uniformly over the receiving surface at the
rate of 15 an hour.
The graduation of a sundial to be placed on a vertical
wall is not difficult, but it is a good little puzzle for testing a
boy's knowledge of the sphere and his powers of visualizing
the true geometrical relations of the parts of a rather com
plicated figure.
Mathematical problems in astronomy are, of course, un
limited, but in school there is no time to touch upon more
than the bare fundamentals.
Whitaker's Almanack is a mine of useful data for problem
purposes.
Stellar Astronomy
The main interest of astronomers, and indeed that of the
general public, is now concerned with the stars and nebulae
rather than with the solar system." With the main facts of
the solar system every boy should be made familiar; but
stellar astronomy is more difficult, the greater part of the
available evidence being merely of an inferential character.
In a very large measure we have to deal with probabilities,
not certainties.
The astronomer's principal instruments are the telescope
(mounted in different ways according to the work to be done),
the spectroscope, the camera, and the interferometer. The
ASTRONOMY 503
lastnamed is outside possible school practice, so is the
camera. But the spectroscope is now in common use in
schools, and as it ranks next to the telescope in the work
of an observatory, its uses should be taught thoroughly.
A course of instruction may be expected to include the
following:
1. Spectrum analysis. Displacement of lines: the causes;
difficulty of interpretation; distance and speed effects con
sidered separately.
2. The galactic system of stars.
3. The extragalactic system: stars and nebulae.
4. Stellar spectra. Interpretation of photographs.
5. Stellar magnitudes, movements, velocities, distances,
temperatures; how determined.
6. Theories of stellar structure: for instance, (i) Edding
ton's, (ii) Jeans'.
7. Solar radiation. Energy and temperature of sun.
Poincare's theorem.
8. Stellar radiation and cosmic radiation generally. Hoff
mann's determination of the sun's contribution to the total
cosmic ultraradiation; inferences therefrom. Hess's views.
9. Relativity. General outline. Einstein's proposed tests.
Confirmation of the tests and final acceptance of the theory.
10. Modern cosmologies: (i) Einstein's, (ii) De Sitter's.
Do they clash? Lemaitre's views how an Einstein universe
may expand to a De Sitter universe.
11. Rival theories as to the future of the universe. British
physicists' views of a universe slowly running down to a state
of thermodynamic equilibrium. Millikan's views of a universe
being continually rebuilt. Evidence pro and con.
How much of this work will be done by the mathe
matical teacher? His task will probably be concerned mainly
with two things: (i) some easy but extremely interesting
arithmetic; (ii) the very difficult subject of Relativity.
Mathematical teachers differ in opinion as to the wisdom
(or folly) of introducing relativity in a Sixth Form course.
504 CRAFTSMANSHIP IN MATHEMATICS
But in view of the farreaching, indeed fundamental, changes
that the subject is bringing about in the whole domain of
physics, it seems desirable that an attempt should be made
to give Sixth Form specialists at least an outline of the subject.
After all, the " special " theory of relativity is easily taught,
and, this done, the much more difficult " general " theory
may be so far touched upon that the final results of the theory
may be fairly well understood by the abler boys. Professor
Rice's and Mr. DurelPs little books may be followed up
by Einstein's own elementary book, and his by Nunn's
Relativity and Gravitation, which is by far the best book
on the subject from the teacher's point of view.*
The arithmetic of stellar astronomy deals with numbers
so vast that it is likely to deceive all but the trained mathe
matician. How, for instance, may we bring home to a boy
the real significance of the following:
1. The sun is losing weight by radiation at the rate of
131 . 10 14 tons a year, yet 2 . 10 years ago it was only
100013 times its present weight.
2. Weight of sun 2 . 10 33 grammes.
3. Temperature of interior of sun = 4 . 10 8 degrees.
4. Number of stars in galactic system = 4 . 10 11 .
5. The 2,000,000 extra galactic nebulae each contain
enough matter to make 2 . 10 9 stars, that is 4 . 10 15 stars in all.
6. The extragalactic nebulae are at an average distance
away of 140 million lightyears (1 lightyear = 6 . 10 12 miles)
and their average distance apart from each other is of the
order of 2 million lightyears.
7. Radius of universe is perhaps 2000 million lightyears
= 2 . 10 9 X 6 . 10 12 miles
= 12 . 10 22 miles.
We shall refer to this subject again in a later Chapter.
Be consistent when using the terms " world ", " uni
verse ", " cosmos ", " space ", " ether ", " spacetime ". It
* For detailed suggestions see Chapter XXXII of Science Teaching.
ASTRONOMY 505
is probably sufficient to tell a boy that the matter containing
universe, no matter how large, is itself within a limitless void.
Do not let him think that the mathematician's convenient
and necessary fiction " spacetime " is any sort of glorified
Christmas pudding mixture. The mathematical partnership
is purely formal. Distinguish between an infinite void and
a limited wavecarrying mattercontaining universe.
In his address to the Mathematical Association, January,
1931, Sir Arthur Eddington said: "About every 1,500,000,000
years the universe will double its radius and its size will
go on expanding in this way in Geometrical Progression for
ever." A rude boy might ask some very awkward questions
on this point, and carry his teacher backwards as well as
forwards in limitless time. It is of no use merely to go back
to an assumed initial state of equilibrium. The boy is certain
to say, and before that?
Books to consult:
In selecting books on Astronomy, don't forget some of the older
writers, e.g. Herschel, Proctor, Lockyer, Ball. Eddington 's, Jeans',
and Turner's books should be known to all teachers of mathematics.
Barlow and Bryan's Elementary Mathematical Astronomy is very
useful. From the teacher's point of view, Sir Richard Gregory's
books take quite the first place. Consult also Dingle's Astrophysics.
Readers who are specially interested in Relativity should read
Dr. John Dougall's searchingly critical article in Vol. X of the
Philosophical Magazine, pp. 81100.
503 CRAFTSMANSHIP IN MATHEMATICS
CHAPTER XXXIV
Geometrical Optics
Present Methods of Teaching often Criticized
We include this subject because it quite properly belongs
to mathematics as well as to physics.
Probably no part of the teaching of mathematics or of
physics is so severely criticized as the teaching of optics, no
matter whether the subject is taught by the mathematics teacher
or by the physics teacher. That there is an urgent need for
some reform will be readily admitted from the discussion
on " The Teaching of Geometrical Optics " that took place
on April 26, 1929, reported fully on pp. 258340 in No. 229
of the Proceedings of the Physical Society. Papers were read
by a number of persons interested in optics, including several
Public School and University teachers and representatives
of the optical industry. A few of the teachers tried to defend
the present system, though not very successfully. The conflict
of opinion centred largely (1), round the place to be given and
the purpose to be assigned, in a teaching course of optics, to
the reciprocal equation (1/u \~ I/v I/f), and (2), round the
question of " rays or waves ". My own quite definite con
clusion from the discussion was that the best way of teaching
the subject is to begin with elementary physical optics in
Forms IV and V, and to defer geometrical optics until Form VI.
Several of the critics found fault with the present system
because it fails to supply a sufficient practical knowledge
of optical instruments and their performance; because pupils
by the end of their course in optics have done little more than
devote their time to elementary algebra and geometrical
diagrams which have but a very slender relation to the subject
under consideration; because, in short, the utility of the subject
is extremely meagre.
GEOMETRICAL OPTICS 507
My own main criticism takes another direction that
the mathematics and the theory of the subject at present
tend to take too early a place in the teaching course, inasmuch
as the physical phenomena underlying the mathematics and
the theoretical arguments have not been studied, the arguments,
therefore, having no real significance.
Rays or Waves?
Hitherto the " ray " method of teaching has been almost
universal in our schools, but the mathematics has been
too much divorced from experiment and its real significance
has been ill understood. In the discussion already referred
to, the method was defended mainly because of its simplicity,
not because of its practical utility. The protagonist of the
wave or curvature method was Dr. Drysdale, for many
years head of the optical department at the Northampton
Institute, London. He advocated the method on the grounds
(amongst others) that (1) it simplifies the teaching; (2) it
harmonizes the teaching of science with optical practice;
and (3) it leads naturally to higher physical optics. The real
advantage of the method seems to be that it places the whole
of optical teaching on a physical basis, and leads naturally
to the study of interference, diffraction, and polarization.
Two wellknown elementary books developing the subject
on a wave basis are those of Mr. W. E. Cross and Mr. C. G.
Vernon.
Whichever method is used, the teacher should be quite
frank in stating that energy can be radiated in two forms,
corpuscles and waves. Both forms are easily illustrated experi
mentally. For example, replicas of diffraction gratings (if
gratings themselves are too expensive to buy) are suitable
for illustrating the periodic character of light. In fact, the
periodic character of light must be experimentally demon
strated in some way before the curvature method can logically
be introduced, and this means a preliminary study of the
velocity of light.
508 CRAFTSMANSHIP IN MATHEMATICS
It is a good thing to teach both methods, and to teach
them more or less in parallel. A ray may, for instance, be
looked upon as a line representing an element of the wave
front, or as a normal to the wavesurface; or the wavefront
may be traced as a series of arcs after the rays have been drawn
graphically.
The best defence of the wave method is that the whole
of physics is, fundamentally, a study of wave systems, and
it is therefore difficult to justify the picking out of one branch
and treating it on an entirely different basis. But the ob
jection to the ray method largely disappears if the ray be
thought of as an element of a wave, and to the lens designer
the ray is the allimportant thing.
Theories of Light
The whole question turns largely on an acceptable theory
of light. But whose theory? Newton's? Fresnel's? Young's?
Maxwell's? Planck's? de Broglie's?
Newton's corpuscular theory failed to account for certain
observed facts. The wave theory which superseded it was
also found to be defective, and to eliminate these defects the
" quantum " theory has been devised. The new theory has
shown that Newton was not wholly wrong in regarding light
as corpuscular, for that theory is based on the experimental
fact that a beam of light may be considered to be broken up
into discrete units called " lightquanta " or " photons ",
" with almost the definiteness with which a shower of rain
may be broken up into drops of water, or a gas into separate
molecules ". At the same time, the light preserves its undu
latory character. Each photon has associated with it a perfectly
definite quantity of the nature of a wavelength.
There seems to be no doubt at all that radiation of all
kinds can appear now as waves, now as particles. But the
fundamental units of matter, electrons and protons, can also
appear now as waves, now as particles. In many circumstances
the behaviour of an electron or proton is found to be too
GEOMETRICAL OPTICS 509
complex to permit of explanation as the motion of a mere
particle, and accordingly physicists have tried to interpret it
as the behaviour of a group of waves, and in so doing have
founded the branch of mathematical physics known as " wave
mechanics ".
In fact it may be fairly said that no single satisfactory
theory of light exists today. The electromagnetic theory
carries us a long way, but in its classical form it is quite
inadequate to carry us the whole way. The powerful methods
devised by Hamilton in geometrical * mechanics and geome
trical optics are being used to found a wavemechanics bearing
to geometrical mechanics a relation similar to that which
waveoptics bears to geometrical optics. The quasi light
particles emerge from this mechanics more or less naturally,
so that we are practically back to Newton and working on
Newton's lines. The two views are blended; neither is
destroyed.
Geometrical optics is as worthy of serious study as geome
trical mechanics. Each is the limiting form when A ~> 0,
and for many purposes this limiting mathematical form is
not only entirely sufficient but it is vastly simpler, mathemati
cally, than the general waveform, whether in optics or in
mechanics. What is not worthy of study (at all events as
physics) is the type of question often set, the solution of which
depends wholly on some mathematical trick. Large numbers
of these are found in such favourite old books as Tait and
Steele's Dynamics, or Parkinson's Optics, or Heath's Geometrical
Optics. Such problems are possibly good as training material
in mathematics, but for the display of mathematical talent
there is an abundance of excellent material that is, in itself,
valuable in physics also.
* I.e. Newtonian.
510 CRAFTSMANSHIP IN MATHEMATICS
The Teacher of Optics
Should optics be taught by the mathematics teacher or
by the physics teacher? Admittedly a mathematics teacher
who has had no training in physics is not likely to be able
to appreciate the natural powers and limitations of optical
instruments, or to grasp the significance of certain matters
in optical theory. Admittedly, too, a physics teacher with
no special knowledge of mathematics will be out of his depth
in the Sixth Form where, in optics, mathematical considerations
count for almost everything, though he will be easily able to
cope with the first considerations of the reciprocal equation,
which, after all, is essentially a natural development of fives
court and billiardtable geometry. There is thus very little
doubt about the answer to our question. The physics teacher
should be responsible for the physical optics in Forms IV and
V, and the mathematics teacher for the geometrical optics
to be done in VI. By geometrical optics is here meant the
really serious mathematical work that should follow the physical
work, work that is partly revisionary but mainly supplementary.
The higher physical work in VI will, however, still have to be
taken by the physics teacher.
Suggested Elementary Course: Mainly Physics
This elementary course is intended to be mainly experi
mental and to be done in the laboratory, all consideration
of the theory of aberration being excluded. Let all serious
mathematics and theoretical developments be postponed to VI.
If the wave method is adopted, wave motion and its sig
nificance will naturally be taught first. Of the many wave
producing machines in the market, select one, and see that
the boys really understand what it teaches. The propagation
of transverse waves may be shown by a ripple tank, illuminated
stroboscopically, so that the apparent rate of propagation may
GEOMETRICAL OPTICS 511
be slowed down. Carry out practical work with real beams of
light, not by pin and parallax methods. The sunbeam offers
a concrete startingpoint.
Devote a lesson or two to showing how fallible the eye
is as a measuring instrument, and why, therefore, instrumental
aids are necessary. Devise experiments to show the limited
power of the eye in unaided vision, and show the capacity
of the eye for distinguishing detail under different conditions
of illumination and size of aperture.
Make beginners familiar with the construction and use
of optical instruments the telescope, the microscope, and
photographic lens. When a boy handles optical instruments,
and learns to adjust, to test, and to use them, he acquires
knowledge of their potentialities and limitations; and he
also becomes acquainted with the language of the subject.
Throughout the course keep in mind elementary notions
both of physiological optics and of the psychology of vision;
also that the eye as an optical instrument is very imperfect,
deceptive, and inconstant. Teach beginners when using
optical instruments the importance of correct illumination;
and the uselessness of increasing magnification beyond the
value suitable for the aperture actually effective in the experi
ment. Show that the apparent brightness of an extended object
cannot be increased by optical means; the moon looks no
brighter through a telescope.
The key to refraction is, of course, the mere retardation
of velocity in a denser medium, and the boys must under
stand clearly that a refractive index is simply a velocity ratio.
The slewing round of the wavefront must be understood
to be just a natural and inevitable consequence of any such
retardation and to be applicable universally and not merely
in connexion with light. The trundling of a garden roller
across a smooth lawn to a rough gravel drive affords a service
able illustration. If the direction of motion across the lawn
is normal to the line of separation between grass and gravel,
there is merely retarded velocity; if oblique, there is a slew
ing round as well.
512 CRAFTSMANSHIP IN MATHEMATICS
Suggested topics:
1. Nature and propagation of light.
2. Waves: motion, length, amplitude, frequency, velo
city.
3. Illumination. Photometry, especially the measurement
of illumination by daylight photometer.*
4. Experiments in brightness, colour, persistence of vision,
fatigue, glare.
5. Reflection and refraction. Concept of the ray as a
line representing the direction of movement of an element
of the wavefront. The use of rays in optical diagrams.
Huygens' principle.
6. Function of lenses; imprinting of curvature.
7. Interference, diffraction, polarization.
8. The spectrum; the spectroscope.
9. The spectrometer: first considerations.
10. The beginnings of mathematics; the reciprocal equation
as a convenient memorandum for elementary work at the
optical bench.
11. Inverse square law; the unit standard source of light,
the unit of luminous flux, the unit of illumination, and their
interrelations.
Suggested Advanced Course: Largely Mathematics
Whatever books on geometrical optics teachers use,
especially if they are old favourites like Parkinson and Heath,
it is a good plan either to compare these with modern standard
works on the technical side of the subject, or to discuss them
with a friend acquainted with the optical industry. The im
portant thing is to find out if the principles laid down in a book
will work.
An examiner reports that at a recent university examination he set a simple
question on the measurement of daylight illumination. Hardly any of the 240
candidates gave a complete answer. A common plan was to balance sunlight
against an electric lamp, using, say, a grease spot, assume the sun to be 93,000,000
miles away, assume the inverse square law, and to calculate the candlepower of the
sun!
GEOMETRICAL OPTICS 513
There is no excuse whatever for teaching the subject
by methods that are out of harmony with applied optics.
Young computers who are taken on at optical works often
find to their disgust that their school and textbook knowledge
is valueless, and they have to be taught anew by technical
experts.
The commonest mistake in optical teaching is due to
the misuse or to the misunderstanding of the sign convention
and notation. This is unaccountable, as the convention
is the result of international agreement. In the optical dis
cussion already referred to, an examiner said that a year or
two ago he marked 250 scripts in the Higher Certificate
examination, the candidates having been taught in schools
in different parts of the country. In the Light paper was a
simple question on a lens, and 247 of the candidates attempted
it, but only 7 of the 247 obtained the correct result. Such a
record of muddleheadedness is utterly inexcusable.
Remember that the basis of all lens work calculation should
be the deviation in a ray at each surface. Suppose that a ray
which diverges from the point A at, say, 15 is to emerge from
the lens system, S, parallel to the axis. Since the whole
deviation is to be 15, and if there are, say, two surfaces,
are the two partial deviations to be 7 each, or in some
other proportion? What are the criteria for what is best?
What are the aberrations? And so on.
After a few simple calculations on a simple lens for actual
wideangle cases, the boys will soon find that the rule given
by Parkinson for the relative radii of the surfaces is by no
(B291) 34
514 CRAFTSMANSHIP IN MATHEMATICS
means always right; it is only right when a ^ 10 (about),
while in many lenses a is very much greater.
Wave optics must not, of course, be forgotten. For instance,
the wave equation and its simple solution should be included.
Suggested topics:
1. Geometrical optics: the reciprocal equation more fully
considered. " Wave " proofs and " ray " proofs compared.
2. The dioptre, sagitta (sag), focal power. Show that the
curvature of a wavefront or surface is measured by the reciprocal
of the radius; the surface with a radius of 1 m. is chosen
as a standard. R dioptres = l/^ met res Exhibit a curve of 1 m.
radius so that the curvature may be visualized. Point out that
for a chord of 895 cm. the curvature in dioptres is represented
by the sag in mm. The application of Euclid, III, 35, to the
sag. The dioptre spherometer.
3. The ideal lens contrasted with the actual lens. (The
solution of problems arising out of actual lenses will in general
be too difficult.)
4. Lenses; spectacles. How the optician is concerned
with the forms of lenses as well as with their power.
5. Combination of lenses with prisms to correct defects
of convergence in the eyes.
6. Thin lenses in contact.
7. Lens combinations.
8. Axial displacement.
9. Chromatic aberration.
10. Spherical aberration: the disc of confusion.
11. Astigmatism, coma, distortion.
12. Photometry further considered. How the distance of
star clusters and spiral nebulae have been determined by the
measurement of the apparent brightness of Cepheid variables
contained therein.
13. Modern instruments; the telephoto lens, range
finders, prism binoculars, kinema projectors.
14. The more elementary considerations of such subjects
as defects of images, collineation between object space and
image space, the optical sine theory, design of instruments.
GEOMETRICAL OPTICS 515
The correction of aberrations by calculation will, in
general, be too difficult; so will the higher order of aberrations
considered by raytracing, though some notion of raytracing
should certainly be given. The general theory of lenses will
also be too difficult. In short, a good deal of this work is
more suitable for the university than for the school. Much
will depend upon the close collaboration of the mathematics
and physics staff. The two aspects of the subjects must be
considered together.
The real value of mathematical work in optics lies in the
discovery of the general principles underlying the actual
behaviour of real optical systems, as contrasted with the imagined
behaviour of ideally perfect systems.
Technical Optics
Few teachers are familiar with technical optics. Very
few have seen even the ordinary operation of grinding a
lens. As for the designing of lenses for special purposes,
or the art of producing optical glass, it is known to very
few persons indeed. Few teachers realize that for ordinary
industrial purposes the index of a glass is not considered
known unless its value is obtained to the fourth decimal
place, and for dispersion to the fifth.
Formerly when an optical system had been designed, the
material prepared by the designer was handed over to a
number of computers expert in the use of logarithmic tables.
But calculating machines are now used, to the operators of
which the computation of the elements, individually and in
combination, of the new optical system is entrusted. These
operators need have no special mathematical equipment,
other than that of a common knowledge of simple trigo
nometrical expressions. Particular rays are traced step by
step through surface after surface for the purpose of de
termining at various stages the longitudinal and transverse
aberrations. These values are assessed by the skilled com
puter, who decides at what particular part of the system a
516 CRAFTSMANSHIP IN MATHEMATICS
modification can best be effected. His special skill is practical,
the outcome of active practice in the industry itself. It
involves, above all, good judgment in the balancing of
one type of aberration against another, for no optical
system can be free from all kinds of aberration. Consider
the amount of work involved in the computation of the
optical system of a typical submarine periscope. Altogether
the number of separate operations is something like 40,000,
the mere recording of which would fill a book of some 250
pages.
Of course all this sort of work is entirely outside anything
that can be done in school, but if a teacher himself is entirely
ignorant of it, how can he help making his subject unreal,
and talking about it in a foreign tongue?
The Sign Convention
Many of the difficulties underlying the teaching of ele
mentary optics in the past have arisen because teachers
have adopted different practices in the use of signs. The
following diagram shows the sign convention that has been
INCIDENT LIGHT ^
agreed upon by the principal optical authorities in the country.
Books to consult:
1. Optics, W. E. Cross.
2. Light, C. G. Vernon.
3. The Theory of Light (new ed.;, T. Preston.
4. Introduction to the Theory of Optics, A. Schuster.
5. Experimental Optics, C. F. C. Searle.
6. Practical Optics, B. K. Johnson.
7. Theory of Optics, P. Drude (trans, by Mann and Millikan).
8. Optics, MullerPouillet, 3rd ed.
GEOMETRICAL OPTICS 517
9. Principles and Methods of Geometrical Optics, J. P. C. Southall.
10. Optical Measuring Instruments, Prof. L. C. Martin.
11. Optical Designing and Computing, Prof. Conrady.
12. Proceedings of the Physical Society, No. 229; the papers by
Mr. T. Smith, Dr. Searle, Dr. Drysdale, Mr. C. G. Vernon, Captain
T. Y. Baker, are all very instructive.
The reader may usefully refer to the memorandum prepared, in
January, 1931, by the Council of the British Optical Instrument
Manufacturers' Association. The facts adduced definitely establish
the preeminence of the British position in the optical industry.
The tests effected in the National Physical Laboratory are alone
sufficient to make that clear.
CHAPTER XXXV
Map Projection
Developable and non Developable Surfaces
It is the geography teacher's business to show how maps
can be outlined on the particular graticule system prepared
for him by the mathematician. This graticule system a
gridiron or latticework system of parallels and meridians
is in its very essence mathematical and should be included
in every school mathematical course.
Fundamental principles of projection will already have
been taught in the lessons on geometry. The principles of
orthographic projection, including socalled " plans and
elevations ", should have been taught thoroughly. It is
just an affair of parallels and perpendiculars, and thence to
the idea of parallel rays of light from an indefinitely distant
source is but a step.
The geometry of the sphere should also be known
thoroughly; for instance, that the area of a circle is ?rR 2 ;
of a sphere, TrD 2 and therefore 4 times one of its great
518 CRAFTSMANSHIP IN MATHEMATICS
circles; of a hemisphere, twice that of its great circle; and
that the volume of a sphere is 7rD 3 /6.
It should be realized that when we look at a sphere we
cannot see the whole of a half of it. The portion of the
visible surface is that encircled by a tangent cone with its
apex at the eye (we neglect binocular vision). A photograph
of a geographical globe would necessarily give a picture of
rather Jess than a hemisphere.
Developable surfaces is another subject that should have
been taught. A paper model of a cube, prism, or pyramid
can be slit open along some of its edges and laid out on the
flat, in other words, " developed ". A cylinder or cone
can be similarly treated. On a cylinder or cone straight lines
can be drawn in certain directions; if the cylinder or the
cone is lying on the table, the line of contact with the table
is one such straight line. But a spherical surface is altogether
of a different type; no straight line can be drawn upon it;
it cannot be developed. A sphere touches a plane at a
point. We cannot cover a sphere with a sheet of paper as
we can a cylinder or cone.
Now the earth is approximately spherical, and any correctly
drawn map is part of that spherical surface. An atlas of
true maps would consist of spherical segments, not flat
sheets. Such an atlas has been made in metal, but it is clumsy
to use and is expensive. For convenience we draw our maps
on the flat, and thus they are all wrong. A map of England
drawn to scale on the surface of an orange would be very
small but large enough for a needle to be thrust through
the orange along a chord from Bournemouth to Berwick.
A perfectly straight tunnel driven between these towns
would pass under Birmingham 4 miles below the surface.
If a map of Europe be sketched to scale on a hollow india
rubber ball, and that portion of the ball be cut out, the portion
has to be stretched a great deal to lie flat, and thus parts
of the map are greatly distorted.
Evidently no map can be drawn on a flat surface accurately.
How do mapmakers set to work?
MAP PROJECTION 519
If we examine an ordinary geographical globe, we see
the equator, the north and south poles, meridians of longitude
running from pole to pole, and diminishing circles of latitude
running " parallel " to the equator. And on this network of
lines we see a true map of the world.
To draw a map, we first draw a network of lines corre
sponding as nearly as possible to those on the surface of the
globe, though they are bound to differ very considerably
from the originals. The network once drawn, we put into
each little compartment, as accurately as we can, the corre
sponding bit of map on the globe. The real trouble is to
draw the network.
An examination of an atlas shows that the various net
works differ much in appearance. Sometimes one or both
sets of lines are straight, sometimes curved, and the cur
vature seems to vary in all sorts of ways. Why? This we
must try to find out.
In an ordinary plan drawn to scale, say of a house or of
a town, we have the simplest form of projection, called the
" orthographic ". To every point in the original corresponds
a definite point in the drawing, and the spatial relations
between the points are faithfully reproduced; only the scale
is changed.
But in a map, the relations may all be changed. There
will, however, still be a systematic onetoone correspondence
of points. Some sort of general resemblance to the original
may always be easily detected, though there is certain to be
distortion of form, or inequality in area, or both.
The mapmaker is bound to sacrifice something. If he
is making a map for teaching geography, he tries to represent
correctly the relative sizes of land and sea areas and thus
provides an equalarea projection. If he is making a map for
a navigator, he tries to show correct directions, and does
not trouble much about size. Or he may be concerned
mainly with correct shapes, and not much with sizes and
directions. Hence he has contrived projections for different
purposes. He has to be content to represent a portion of
520 CRAFTSMANSHIP IN MATHEMATICS
the earth's surface accurately in certain respects and to
let other considerations go.
The plan adopted is to project the curved lines from the
globe on to (1) a plane surface, or (2) a developable surface
(cylinder or cone).
Some projections are readily effected geometrically; they
are easy to draw and to understand. Other projections are
not strictly geometrical: they are compromises, effected for
some particular purpose, and are often called transformations.
In these cases pointtopoint correspondence is determined
merely by formulae which express the position of each point
on the plane of the projection in terms of the position of the
point on the spherical surface to which it corresponds.
Projection Shadows
It is possible to obtain geometrical projections by casting
shadows. A light is placed in a suitable position, and a pencil
outline of the shadow of the globe is traced on a conveniently
placed plane. This done, it is easy to see how a better pro
jection may be made with ruler and compasses.
Of course if we use a solid globe the shadow will be
merely a black circle. We require a hollow translucent
globe, with the meridians and parallels painted black on the
surface, and a strong light inside. If the globe is fixed near
a sheet of white paper on the wall or on the table, the shadows
of some of the meridians and parallels will be cast on the
paper, and those fairly near the globe will be clear enough
to be pencilled over. A large white porcelain globe used for
gas and electric lighting answers the purpose well.
When teaching 40 years ago, I found that a better plan
was to use a spherical wire cage instead of a globe, made
something after the pattern of the oldfashioned wire pro
tectors of naked gasflames in factories. Such a cage 2' or
2' 6" in diameter is easily made in the school workshop.
It is merely a question of bending wire and soldering a number
of joints. For the equator, a rather stouter wire should be
MAP PROJECTION
521
used than for the other circles. The meridians are best
not made of complete circles but of rather less than half
circles, fastened into a ring 4" or 5" in diameter, after the
manner of the ribs at the top of an umbrella. It is true that
the actual north and south poles will be missing but this
cannot be helped, the crossing of 12 wire circles at a common
point not being practicable. The meridians and parallels
may be placed at 15 intervals. Two halfcages are also
desirable, one with a pole at its centre, one with a point on
Fig. 275
the equator at its centre. The three should be mounted on
suitable stands, in order that, in use, they may easily be
kept in a fixed position.
The main difficulty is the provision of a suitable light.
Theoretically we require the light to be concentrated at a
point. As this is impossible, we use a small electric bulb,
porcelain or similar material, with the most powerful light
obtainable. A darkened room is, of course, necessary.
Main Types of Projection
The principal projections may be grouped under six
main heads: (1) zenithal or azimuthal; (2) globular; (3)
conical; (4) cylindrical; (5) sinusoidal; (6) elliptical. Of
most of these there are various modifications.
522 CRAFTSMANSHIP IN MATHEMATICS
(1) Zenithal or Azimuthal Projection
This projection derives its two names from the facts
(1) the map is symmetrical about its central point, just as
the stellar vault is symmetrical about the zenith of the observer;
(2) the projection preserves the azirmiths of distances measured
from the map's centre.
There are three distinct types of this projection: (1)
orthographic; (2) stereographies (3) gnomonic. (See fig. 276.)
1. Orthographic. This is simply an affair of perpen
diculars and parallels. As we cannot obtain parallel rays
by artificial light, we must use the sunlight at some con
venient hour. Let the paper prepared to receive the shadow
be placed at right angles to the direction of the sun's rays.
Fix the skeleton wire hemisphere (ii) so that the equator
is parallel to the paper; the parallels of latitude will be
projected as circles of true size, the meridians as radii of
these circles. Geometrically, we draw the projection exactly
as in geometry. Note that the scale along the circles is always
true, but that the radial lines are foreshortened more and
more as the distance from the centre increases.
, 2. Stereographic. Set up the same wire hemisphere,
with its equatorial plane vertical and parallel to the projection
plane, and place the lamp at the further extremity of the
diameter corresponding to the earth's axis, that is at the
" south pole ". The parallels of latitude are again projected
as circles, but they are enlarged, the equator being twice
the size of the original. The meridians are projected as
radii, as before.
This projection has a general similarity to the orthographic,
and its geometrical construction is a useful exercise. The
scale is increased equally along the meridians and parallels,
and some good Sixth Form problems may be based on the
projection. In particular, the projection provides a ready means
of studying the sum of the angles of a spherical triangle.
3. Gnomonic. For this projection, the light is placed at
the centre of the sphere. Although the parallels of latitude are
MAP PROJECTION
523
o
o
o
z
o
O
u
a:
u
fe
o
x
a
o
i
oc
O
524
CRAFTSMANSHIP IN MATHEMATICS
still projected as true circles, they are still more enlarged,
and the equator itself, having the light in its own plane,
cannot be projected at all. The general appearance of the
projection is similar to that of the other two, but obviously
the areas very far from the pole are greatly distorted in the
projection. The figure shows the polar region to within
30 of the equator. It makes a fairly good map for areas
within 30 of the pole.
The three projections may be usefully compared in this
way:
N M P Q
Let NESQ' be a meridian of the earth, and XY its pro
jection plane. Take a point L at polar distance a. Then
angle LSN a. Let radius be R.
Orthographic projection of arc NL = NM = R sin a.
Stereographic = NP = 2R tan^a.
Gnomonic = NQ = R tana.
Observe that in the orthographic projection the outer
circles are crowded together, in the Stereographic the outer
are farther apart than the inner, and in the gnomonic the
outer circles get so far apart as to be useless. It is some
times convenient to arrange these circles at equal dis
MAP PROJECTION
525
tances apart, and then we have the zenithal equidistant pro
jection. It is also possible for the distances of the parallels
of latitude so to be regulated that the area enclosed by any
parallel is equal to the area of the globe cut off by the same
parallel, and then we have the zenithal equalarea projection.
Strictly, these are not true projections, but the associated
geometry is interesting and instructive.
GO;
(2) Globular Projection
All three zenithal projections are sometimes called " per
spective " projections, since they can be cast as shadows.
But the globular projection
cannot be cast as a shadow,
and is therefore nonperspec
tive.
The geometry is a useful
exercise for beginners. The
projection is commonly used
for maps of the world in two
hemispheres. The figure repre
sents one hemisphere. Divide
the equatorial diameter into
an equal number of parts, say
parts representing 30. Divide
the circumference similarly.
The curves are all arcs of circles, each to be drawn through
three points. The mathematics of the projection is of the
simplest.
(3) Conical Projection
For this we require the wire skeleton of the complete
globe, with the light fixed at the centre. The shadow will
be cast, not on a plane, but on the inner surface of a white
paper cone.
Fold up, in the usual way, a common filter paper, and
526
CRAFTSMANSHIP IN MATHEMATICS
fit it into a funnel. It makes a cone with a 60 apex. A half
circle of paper would make the same cone, the two halves
of the diameter being brought together. A sector having
an apex of less than 180 folds up into a more pointed cone;
one with an apex of more than 180 folds up into a flatter
cone. A sector of 360 (a complete circle) necessarily remains
a plane.
Make a white paper cone (of about 130 apex in the flat),
slip it over the polar region of the skeleton globe so that the
apex is in a line with the axis of the globe. The cone touches
the sphere tangentially, viz. in a circle, and this circle is a
parallel of latitude. If this corresponds with one of the wire
Fig. 279
circles so much the better. Now gently mark in the out
lines of the cast shadow. This is pretty easy in the neigh
bourhood of the line just mentioned, but the cast shadow
gets very faint as we get farther away from the line. Now open
out the cone on the flat (fig. 279, ii), and we have an ordinary
conical projection. The arc represented by a heavy line
RSR' is the circle of contact RS in (i), the " standard parallel ",
and it is divided exactly as the circle it touches on the sphere
is divided.
The solid angle N of the sphere = 4 right angles. The
angle of the cone when developed is angle RP'R'. The
ratio of the latter angle to the former is called the constant
MAP PROJECTION
527
Fig. 280
of the cone. It is a simple Fifth Form problem to prove that
this constant is the sine of the latitude of the standard
parallel.
The geometrical construction is simple. Observe that
the parallels are arcs of circles, and that the meridians are
straight lines. Since meridians are great circles and their
planes pass through the centre
of the globe, these planes must
bisect the cone and therefore
cut its surface in straight lines.
The projection is commonly
used for countries in middle
latitudes if the latitude is not
of too great an extent, e.g. for
England. The conical projec
tion with two standard parallels
(fig. 280) is a common projection
for the larger European coun
tries. Its principle is equally
simple.
(4) Cylindrical Projection
Take a large sheet of white paper and convert it into a
cylinder of the same diameter as the skeleton wire sphere.
Its length should be 3 or 4 times the diameter. Slip it over
the sphere so that the equator is in about midposition,
and place the light at the centre of the sphere. A shadow
of a part of the wire sphere is cast on the cylinder. Obviously
the shadows of the two poles cannot be cast on the cylinder
at all, and high latitudes are cast at great distances, with
consequently great distortion. The small circle of latitude
AB will appear as A'B'; in fact all circles of latitude will
be projected as circles on the cylinder and will all be of the
same size as the equator. All meridians, being great circles,
will be cast as straight lines. Open out the cylinder on the
flat (ii), and the projection is seen to consist of a net of rect
angles. E'Q' = TrEQ and may be subdivided in the usual way
528
CRAFTSMANSHIP IN MATHEMATICS
The projection is not of much practical value. Except
in the immediate neighbourhood of the equator there is far
too much distortion.
Fig. 281
But various modifications of this primary cylindrical
projection have been adopted, two of them being noteworthy:
(1) Lambert's equalarea projection, and (2) Mercator's
projection.
1. Lambert's projection. Construction: divide the quadrant
Fig. 282
NQ (fig. 282) into an equal number of parts, say 6 of 15
each, and draw parallels to EQ, and so obtain parallels of
latitude. For meridians, make E'Q' TrEQ, and divide
up into intervals of, say, 30. Note that the parallels of lati
MAP PROJECTION 529
tude are horizontal lines at a distance of r sin A from the
equator (A = lat.).
It is a wellknown theorem in geometry that the area
between any two parallels on the enveloping cylinder is
equal to that of the corresponding zone on the globe. Hence
the area of the rectangle MM' is equal to the area of the globe.
The proof of the theorem should be given.
2. Mercator's orthomorphic projection. This is the best
known of all projections; it is used for navigation purposes,
and for maps of the world. But it is responsible for many
geographical misconceptions, for instance the misleading
appearance of the polar areas, which are greatly exaggerated.
Greenland is made to appear larger than South America,
though only onetenth its size.
As with all cylindrical projections, the meridians are
equidistant parallel lines; the parallels of latitude, on the
other hand, increase in distance from one another the farther
they are from the equator. This spacing of the parallels
of latitudes is so arranged that at any point of intersection
of parallels and meridians (in practice, any small area), the
scale in all directions is the same. Hence the projection is
orthomorphic. Literally the term means " preserving the
correct shape ".
The essential characteristic of the projection, then, is this
that at any point the scale along meridian and parallel is
the same. We give Dr. W. Garnett's ingenious illustration
of the method of effecting this.
Dr. Garnett takes a very narrow gore, i.e. a strip between
two meridians on the globe (cf. the surface of a natural
division of an orange, selected for its narrowness), and
spreads it out as flat as possible; if very narrow there is no
great difficulty in spreading it out very nearly flat, without
much distortion; then it is very nearly an equalarea strip,
i.e. its area on the flat is very nearly the same as when it was
part of the curved surface of the sphere. The length of the
spreadout gore is, of course, half the circumference of the
orange.
(E291 35
530
CRAFTSMANSHIP IN MATHEMATICS
Let NAB represent the half gore, AB representing 10
at the equator; and let NM be the central meridian.
Divide NM into 9 equal parts, and through the points of
division draw the parallels shown in the figure; these re
present 10 intervals of latitude from the equator to the
pole. Suppose the gore to be made of malleable metal.
30' 40' 60
Fig. 283 *
Hammer it out in such a way as to cause it to spread to the
uniform width AB. Clearly we cannot do this in the im
mediate neighbourhood of N: there would not be enough
metal. Hence cut the gore off at about 85. But the gore
cannot be hammered out without expanding in length as
well as in breadth. At, say, 40 little hammering will be
required, and the additional length there will be slight;
but at, say, 70 much hammering will be required to produce
Fig. 284
the necessary additional width, and therefore there will be
much additional length produced. At 45 the ratio of the
increased width to the original width is \/2 : 1, and therefore
the length of the strip at 45 is increased \/2 times, and hence
the area is increased there <\/2 X 1/2 times, that is, twice,
and the thickness is therefore halved. At 60 there will be a
doubling of width and therefore a doubling of length, i.e.
* Figs. 283 and 284 are made to lie down, to save space. Normally, the gores
would be given an upright position.
MAP PROJECTION 531
the area will be multiplied by 4 and the thickness reduced
toj.
It is easy to imagine the whole series of 36 gores (fig. 284
shows 4) placed side by side, and rolled out until the edges
meet and 36 rectangles are formed.
Generally, every little strip parallel to the equator is
increased both in length and breadth in proportion as the
radius of the sphere is to the radius of the circle of latitude
where the strip is situated. At 80 the area is increased
about 33 times, and at 85 about 132 times. The figure
(fig. 283) shows roughly how the gore between and 80
is hammered out into the rectangle ABPQ.
If the 36 gores were extended to lat. 80 N. and S., and
placed side by side, we should have a rectangle 36 times
AB in length and twice AP in height, and we should have the
framework for a Mcrcator map of the world between the
parallels 80 N. and 80 S.
The point about the whole projection is the retention of
true shape, though this applies to only very small areas.
At the equator, areas are unchanged; at 80 they are in
creased 33 times.
The shapes of small areas are magnified, not distorted.
Strictly the orthomorphism is applicable only to points and
is therefore only theoretical.
Construction of a M creator map. The radius of a parallel
of latitude on a sphere of radius r is r cos#. Hence if a degree
of longitude in latitude is to be made equal to a degree
at the equator, its length must be divided by cos0. If the
scale of the map is to be increased in all directions in the
same ratio, then the length of the degree of latitude measured
along the meridian must also be increased in the same ratio.
If y be the distance of the parallel of latitude from the
equator in the Mercator map of a sphere of radius r,
e
a formula which may be evaluated by Sixth Form boys.
532 CRAFTSMANSHIP IN MATHEMATICS
The distances of the parallels from the equator are, in
terms of the radius, approximately, for
10
176 R
50
1011 R
20
356 R
60
1317R
30
55 R
70
1736 R
40
763 R
80
2436 R
These values should be checked from a Mercator in a good
atlas: equator = 27T.R.
Mercator, and Great Circle Sailing. The special merit
of Mercator's projection lies in the fact that any given uniform
compass course is represented by a straight line. All meridians
are exactly north and south, and all parallels exactly east
and west. Hence a navigator has only to draw a straight
line between his two ports, and the angle this line makes
with the meridian on the map gives his true course for the
whole voyage.
Any straight line drawn in any direction on a Mercator
is called a rhumb line\ it crosses all parallels at a constant
angle, and all meridians similarly. A sailor who is told to
sail on a constant bearing simply sets his compass according
to the rhumb line.
But this course may not be the shortest; it cannot be,
unless it is along the equator or along a meridian, i.e. along
a great circle. A rhumb course in any other direction is not
along a great circle, and we know that the shortest distance
between two points in a sphere is along the great circle
passing through them. Economy makes the navigator take
the shortest course if he can. How is he to find it?
A rough and ready way would be to take a wire hoop
that would exactly fit round the equator or round one of the
meridians (and therefore round a great circle: we neglect
the ellipticity of the earth), hold it over the globe so that it
passed through the two ports at the ends of the course under
consideration, chalk in the curve, and then transfer the
curve to the Mercator, freehand, as accurately as the corre
sponding graticules would allow.
MAP PROJECTION
533
A navigator always follows a great circle if he can, not
the rhumb line, and for his special use great circle courses are
calculated and laid down on a Mercator's chart.
If ARE is the rhumb line between two places A and B
(the figure is a fragment of a Mercator chart), and AGCB is
the great circle (and therefore shorter than the rhumb line),
a navigator might sail along a series of chords AG, GC, CB,
altering his course at G and C. He would not quite follow
Fig. 285
the great circle, but he would follow a much shorter route
than the rhumb line course.
Give the boys examples of the course between, say,
Japan and Cape Horn, Plymouth and New Orleans, Cape
Town and Adelaide. Let them mark in roughly both the
rhumb line and the great circle courses on a Mercator chart.
Remind them of the deceptive geometry, as in fig. 285, where
the chord represents a longer distance than the arc it subtends.
To trace the course of a great circle on a Mercator chart.
Any great circle must cut the equator at two places and at
a given angle. Hence it will cut (i) a given meridian at a
point whose latitude can be determined, and (ii) a given
parallel of latitude at a point whose longitude can be de
termined.
Assume that we are given:
(i) a, the inclination of the great circle to the plane of
the equator;
(ii) A, the longitude, measured from one of the points
of section, of a meridian in latitude L.
534 CRAFTSMANSHIP IN MATHEMATICS
Then the following equation may be established:
tanL = tana . sin A,
or, sin A cot a .tanL.
From this equation, either the latitude can be determined at
which the great circle cuts any meridian, or the longitude
at which it cuts any parallel. The equation may therefore
be used to trace the course of a great circle on a Mercator
chart. Sixth Form boys should work through a few of the
exercises in Nunn, Exercises, Vol. II.
Aviators are naturally much interested in great circle
sailing. Let the boys determine an aviator's route between
two given places, say 5000 miles apart, by stretching a string
over a geographical globe. Then ask them how an aviator
would set his compass. Let them lay down the course on
a Mercator chart (graphically and approximately will do),
and see how it differs from the rhumb line, and how compass
directions might be determined by a succession of chords.
(5) Sinusoidal Equal area Projection
This is sometimes called the Sanson Flamsteed projection;
it is used mainly for world maps. An equalarea or " homo
lographic " projection is a projection where shape is sacrificed
to equality of area.
It differs widely from the geometrical and (mainly)
shadow projections already considered.
The equator (= 2?rR) is true to scale. The central meridian
( TrR) is also true to scale. Parallels of latitude are equidistant
horizontal lines. All the meridians are of the form of sine
curves. Each parallel is equally divided by the meridians,
which are nearer and nearer together towards the poles.
Fig. 286 shows a quarter of the complete projection of the
world map; EZ = 2NZ. Divide NZ into, say, 6 equal parts
(of 15 each), and EZ into 6 parts (of 30 each). Each hori
zontal straight line is equal in length to the corresponding
MAP PROJECTION
535
circle of latitude. Through the extremities of these lines
draw the curve EN which represents the boundary of the
quarter map. Divide every parallel into 6 equal parts, similar
to EZ, and draw curves through the corresponding points
N
60
30"
71
180
60'
Fig. 286
of division; these curves are meridians (in the figure, half
meridians).
The disadvantage of the projection is that towards the
edge the meridians are very oblique and thus the shape is
much distorted. The graticules along the equator and central
meridian, on the other hand, practically retain their original
shape.
In any projection graticule, the horizontal lengths are
exactly the same as in the graticule on the globe; and the
vertical height of the projection graticule is equal to the
length of the corresponding piece of meridian on the globe.
Hence the area of any projection graticule is equal to the
area of the corresponding graticule on the globe, or the whole
area of the map is equal to the whole area of the globe.
Each curved meridian is a sine curve: why? Might the
sine curves be drawn before the parallels?
The projection is very good for maps of Africa and South
America. Why?
536
CRAFTSMANSHIP IN MATHEMATICS
(6) Mollweide's Elliptical Projection
This projection is also used for world maps. Again the
parallels are horizontal lines. The meridians are ellipses
(there are two special cases: the central meridian is a straight
line and the 90 meridian is a circle).
Again the area of the map is equal to the area of the surface
of the globe.
Since the area of the surface of the sphere is equal to
4 times the area of its great circle, the area of the hemisphere
is equal to twice the area of its circular base.
Fig. 287
Let the radius of the globe, fig. 287 (i), be R. Draw a circle
(ii) of radius \/2.R (= CB). Area 2?rR 2 sq. in., which is the
area of the half globe. Let C be the centre of the circle;
draw a horizontal diameter ACB and a vertical diameter
NCS. Produce AB so that CE = 2CA and CQ = 2CB.
Draw an ellipse having EQ and NS for axes. It is one of the
properties of the ellipse that if any line KLMN' be drawn
parallel to EQ cutting the ellipse and the circle, KN' = 2LM;
and as this is true for any such line, it follows that the area
of the ellipse = twice the area of the circle = the area of
the globe.
Divide EQ into equal parts and through the points of
division draw ellipses with NS as a common axis; these are
MAP PROJECTION 537
the meridians. Evidently all gores (e.g. NnSC, NwSw) are
equal in area. For an equalarea projection, it remains to
divide these gores by parallels of latitude into the same areas
as the corresponding gores between the meridians on the globe
are divided. This is the only difficult part of the problem.
We have to draw KN' so that it will correspond to some
particular degree of latitude <f> on the globe. Fig. 287 (i) repre
sents a section of the globe through the great circle NA'SB'.
In fig. (ii), the circle represents the area of the hemisphere
and the ellipse the area of the whole sphere.
The area of the zone L'A'B'M' on the spherical surface
(radius = R) in fig. (i) is equal to twice the area of the zone
LABM on the plane surface (radius ^/2.R) in fig. (ii).
We have to find the angle MCB. Let it equal a. Then
2a f sin2a = TT sin9.
It is not easy from this equation to obtain a in terms of <,
but it is quite easy to determine cf> in terms of a. Hence if
any parallel be drawn in the ellipse, and the angle a is measured,
the latitude </> to which it corresponds is found at once.
The formula should be established by the Sixth Form.
Choice of Projection
Let the boys examine a good modern atlas in which the
projections used are named; and get the boys to discover
why a particular projection is used in each case. This may
give rise to an interesting discussion.
Books to consult:
1. A Little Book on Map Projection, Garnett.
2. The Study of Map Projection, Steers.
3. Map Projections, Hinks.
538 CRAFTSMANSHIP IN MATHEMATICS
CHAPTER XXXVI
Statistics
The Importance of the Subject
It is highly desirable that an elementary study of this
subject shall be included in any Sixth Form course. Statistics
enter largely into modern science and administrative practice,
and the underlying principles have now been so well worked
out and have become so definite, that no large office, govern
ment or local , can afford to be without at least one well
trained statistician. The newer developments of psychology
depend almost entirely upon a rational interpretation of
statistics. There are some teachers who are still ignorant
of the principles underlying the correct handling of the
statistics of everyday school practice; and thus they are
necessarily unable to make the most effective use of, for
instance, an ordinary sheet of tabulated examination results.
I have seen the subject taken up seriously in only two
or three schools, and have therefore had little experience of
the methods of teaching it. The teaching suggestions in
Professor Nunn's Algebra are recognized as the most practi
cable yet made, and the topics he selects for inclusion in a
school course seem to be just about right. The technical
side of the subject is, of course, rather difficult for boys,
but the fundamentals are easy to grasp, and it is possible
to map out an excellent preliminary course that will give a
good general insight into the subject and into its methods.
The main problems to be considered may be grouped
under the three usual heads: (1) frequency distribution of
a series of measurements or other statistics; (2) frequency
calculation: probability; (3) correlation.
STATISTICS 539
Frequency Distribution
Frequency distribution is concerned with the best ways
of recording statistics and of expressing most simply and
effectively the information which they contain. Suppose,
for instance, a Local Education Authority has examined
20,000 children between the ages of 10 years 6 months and
11 years 6 months for scholarships to be held in the local
Secondary Schools. What would be the best way of recording
the results, so that not only might their significance and its
implications be readily apprehended but also that the record
might form a simple means of comparison with similar
records elsewhere?
Suppose that all the examination papers were arranged in
20 piles, according to the percentage of marks awarded to
each paper, 0% to 5%, 5% to 10%, and so on up to 95% to
100%. The height of the piles would exhibit to the eye the
frequency distribution of the marks, the number of papers in a
pile giving the frequency of the particular mark in that pile.
A logically setout record of the whole of the results might
be called a frequency table. A column graph showing the
number of papers in each pile would afford a useful alternative
means of exhibiting the frequency distribution; such a fre
quency diagram is commonly called a histogram.
The first thing to do is to familiarize the pupils with the
main forms of frequency distribution in tables and diagrams.
The records in Government " Blue Books " are often useful
in this connexion.
A frequency diagram differs from the ordinary, graph of
algebra and physics. The latter represents the relation between
two variables, or the values of a function which corre
spond to different values of a single variable. But a frequency
diagram serves simply to show how often each value of the
variable is met with in some record.
Examples of the forms of frequency distribution may be
drawn from very different sources from anthropometric
measurements, from economics, from meteorology, from
540
CRAFTSMANSHIP IN MATHEMATICS
medical records, and from the records of the workings of
what (in our ignorance) we call chance. The various forms
display resemblances that are often most pronounced when
the diversity of origin would seem to be greatest. The re
semblances are brought out very clearly by the frequency curves.
The pupils should learn that the smooth curve really
represents the interpretation of the ideal distribution to which
actual samples might be expected to approach if they contained
a sufficient number of cases drawn from a field sufficiently
wide to be really representative. When the curve is drawn
with this idea in view, it is always of the same general pattern,
\
(0
Fig. 28
more or less bellshaped. The curves are, however, easily
sorted out into seven distinct types, two of them symmetrical
and five of them asymmetrical or skew.
Fig. 288 (i) is an example of an almost perfect symmetrical
curve, imposed on its histogram, which, however, is itself
unlikely to be quite symmetrical; it is the " normal " type of
curve. Fig. 288 (ii) is an example of a skew curve also imposed
on its histogram. If a curve evaluated from an ordinary
examination marksheet was very skew, i.e. varied con
siderably from the normal, it would suggest that an in
quiry was necessary.
The normal curve was formerly spoken of as the graphic
representation of the " law of error ", it being thought,
perhaps naturally, that the mean of the distribution was the
number (physical measurements in human beings, for in
stance) which represented nature's intention, deviations there
STATISTICS 541
from being " errors ". But it is now recognized that ordinary
distributions are not, even ideally, normal, and that skew
ness in them is almost inevitable, though probably in most
cases the skewness or asymmetry is moderate.
Enough should be done to teach the boys the fundamental
fact that although when the data are few the columns which
represent the frequencies of occurrence may exhibit no
orderly arrangement, yet order invariably appears as the
data become sufficiently numerous.
There will seldom be time for pupils to deal with the
formulae representing statistical graphs, though such work is
certainly interesting and valuable. But the pupils should
be made thoroughly familiar with the significance of the
ordinary statistical phraseology. The mean is the ordinary
arithmetical mean and is easily found when all the in
dividual measurements are given. The median is the middle
measurement; it is the measurement corresponding to the
ordinate which bisects the area of the curve; it may be
found by calculation based upon the known properties of
the curve, or (roughly) by using squared paper and counting
up the squares under the curve. If we examine 5 boys and
their percentage marks are, respectively, 85, 80, 60, 55, and
30, the mean (average) mark is 62, and the median (middle)
mark is 60. The lower and upper quartiles are the two measure
ments onefourth from the beginning and onefourth from
the end of the whole series. The interquartile range includes
the middle half of the whole series. The mode is the measure
ment corresponding to the highest ordinate of the frequency
curve drawn over the histogram.
Deviation. A boy may obtain high marks or low marks;
a man may be tall or short; in both cases there is deviation
from a standard. In fact, in any series there is bound to be
" dispersion ". To measure the deviation, what standard
should be taken?
One recognized way of indicating the dispersion of a set
of measurements, when something more concise than a
frequency table is required, is to state the interquartile range,
542 CRAFTSMANSHIP IN MATHEMATICS
or, more usually, the semiinterquartile range, of the statistics;
only the middle half of the measured cases is considered. This
semiinterquartile range is often called the quartile deviation.
Or we may strike an average of the arithmetical differences
between the various measurements and some selected standard
measurement, e.g. the median, the mean, or the mode. Since
the sum of the differences of the measurements from the
median is less than from any other standard ordinate, the
" mean deviation " of a set of measurements is, as a rule,
calculated with reference to their median.
But the most useful measure of dispersion is that which
takes account not of the deviations themselves but of their
squares. Squaring the deviations gives greater weight to
the larger ones, and has the mathematical advantage of making
the numbers positive. Take the deviations, square them,
find the mean of the squares, and then take the root of the
mean: this gives the useful measure of dispersion called
the standard deviation.
Whether the subject is taught or not, the mathematical
staff would find it a great advantage to graph their periodical
examination results statistically. What does the graph teach?
What does its skewness teach? What can be learnt from the
interquartile range? What can be learnt from a comparison
of the quartile, mean, and standard deviations? Which is
the more useful mark in a marksheet, the mean or the
median? Why? And so on.
The fact should be impressed upon learners that the
graphic method of presenting statistical data has advantages
over the tabular statement of the same data, though it is
necessary to remember that the facts cannot be more
accurately represented by a diagram than by the data from
which the diagram is constructed. Indeed, the diagram may,
from imperfect draftsmanship, fall short in accuracy of the
statistical tables it represents. The graphical presentation
has, however, the important advantage that it presents
lengthy series of data in a form in which the majority of users
STATISTICS 543
of them find it easier to grasp their sequence and their re
lations than when presented in the form of tables.
Frequency determined by Calculation
The previous section dealt with the analysis of frequencies
actually given. It is now necessary to refer to the possibility of
predicting them among events that have never been observed. It
is in connexion with this problem that the topics, (1) combi
nations and permutations, and (2) probability, are best treated.
The calculation of probabilities is nothing more than the
calculation of frequencies. Probability is not an attribute
of any particular event happening on any particular occasion.
It can only be predicted of an event happening, or conceived
as happening, on a very large number of " occasions ", or
of an event " on the average ", or " in the long run ". Unless
an event can happen, or be conceived to happen, a great
many times, there is no sense in speaking of its probability.
Frequency would be a better word than probability in the
study of the subject generally, but the latter word has become
definitely established.
Make sure that the boys understand the notion of " in
dependent events ". The fall of a tossed coin is an independent
event. Whether it will fall " head " or " tail " the next time
it is thrown depends not at all on how it fell last time, or
the last thousand times. If, for example, there had been
a run of a hundred heads, the " chance " that the next
throw would also be a head is just as great as before.
Another idea the pupils must grasp is that frequency
predictions are possible only in so far as the events predicted
can be regarded as compounded of independent elementary
events whose characteristic behaviour is already known.
Thus, knowing that the spin of a coin is an independent
event which will, in the long run, turn out heads and tails
with equal frequency, we can predict with confidence what
will happen (again in the long run) in the case of an event
which consists in the tossing of (say) 10 coins.
544 CRAFTSMANSHIP IN MATHEMATICS
1. Combinations and Permutations. Apart from the
" tricky " problems that occur in some of the textbooks
(they are of no importance and may be ignored) this subject
seems to be taught well. Most mathematical teachers seem
to have neat little devices for working out nPr y nCr, &c.,
from first principles, it may be by ringing the changes syste
matically, and neatly classifying the results, for a group of a
few letters on the blackboard. Even boys of average ability
soon get to like the little stockproblems about people sitting
round a table, or about the selection of elevens. Do not spend
much time on the subject; it is not worth while. But give
plenty of oral practice in such exercises as finding the value
of 10 C 3 , 10 C 7 , &c., and do not forget the evaluation of co
efficients in expansions.
2. Probability. This is a more serious topic, though its
more elementary considerations are easily within the range
of school work.
Justification for teaching the subject is hardly necessary.
It is by far the best application of the theory of permutations
and combinations, but much more than that, it enters into
the regulation of some of the most practical concerns of
modern life, for instance in the use of mortality tables,
insurance and annuity problems, and so forth. The following
arguments and examples * may serve as a suitable introduc
tion to the subject.
When we say that the probability that an event will
happen in a certain way is I/;/, what we mean is that the
relative amounts of knowledge and ignorance we possess as to
the conditions of the event justify the amount of expectation.
The event itself will happen in some one definite way, exactly
determined by causation; the probability does not determine
that, but only our subjective expectation of it. It is from
this combination of knowledge and ignorance that the cal
culation of probability starts.
Fundamentally, the theory of probability consists in
* Scientific Method, pp. 260 seq.
STATISTICS 545
putting similar cases on an equality, and distributing equally
among them whatever knowledge we possess. Throw a
penny into the air, and consider what we know in regard
to its way of falling. We know that it will certainly fall upon
a side, so that either head or tail will be uppermost; but
as to whether it will be head or tail, our knowledge is equally
divided. Whatever we know concerning head, we know also
concerning tail, so that we have no reason for expecting
one more than the other. The least predominance of belief
to either side would be irrational; it would consist in treat
ing unequally things of which our knowledge is equal. We
must treat equals equally.
The theory does not require that we should first ascertain
by experiment the equal facility of the events we are con
sidering. The more completely we could ascertain and
measure the causes in operation, the more would the events
be removed from the sphere of probability. The theory
comes into play where ignorance begins, and the knowledge
we possess requires to be distributed over many cases. Nor
does the theory show that the coin will fall as often on the
one side as the other. It is almost impossible that this should
happen, because some inequality in the form of the coin,
or some uniform manner in throwing it up, is almost sure
to occasion a slight preponderance in one direction. But
as we do not previously know in which way a preponderance
will exist, we have no reason for expecting head more than
tail.
Suppose that, of certain events, we know that some one
will certainly happen, and that nothing in the constitution
of things determines one rather than another; in that case,
each will recur, in the long run, with a frequency in the
proportion of one to the whole. Every second throw of a
coin, for example, will, in the long run, give heads. Every
sixth throw of a die will, in the long run, give ace.
The method which we employ in the theory consists in
calculating the number of all the cases or events concerning
which our knowledge is equal.
(E291) 36
546 CRAFTSMANSHIP IN MATHEMATICS
Let us suppose that an event may happen in three ways
and fail in two ways, and that all these ways are equally
likely to occur. Clearly, in the long run, the event must
happen three times and fail two times out of every five cases.
The probability of its happening is therefore j?, and of its
failing, f . Thus the probability of an event is the ratio of
the number of times in which the event occurs, in the long
run, to the sum of the number of times in which the events
of that description occur and in which they fail to occur.
An event must either happen or fail. Hence the sum of the
probabilities of its happening or failing is certainty. We
therefore represent certainty by unity.
The usual algebraic definition of probability is as follows.
If an event may happen in a ways and fail in b ways, and all
these ways are equally likely to occur, the probability of its
happening is  , and the probability of its failing is
j a + b
 , (In mathematical works, the word " chance " is
a + b
often used as synonymous with probability.)
It should be noticed that  +   1; also that
, a f b a + b
1  =  . Thus, if p be the probability of the
a } b a + b
happening of an event, the probability of its not happening
is 1 p.
When the probability of the happening of an event is
to the probability of its failure as a is to i, the odds are said
to be a to b for the event, or b to a against it, according as
a is greater or less than 6.
Suppose that 2 white, 3 black, and 4 red balls are thrown
promiscuously into a bag, and a person draws out one of
them, the probability that this will be a white ball is , a
black ball, i], and a red ball, <J.
A few simple problems will help to illustrate the prin
ciples involved.
1. What is the probability of throwing 2 with an ordinary
STATISTICS 547
die? Any one face is as likely to be exposed as any other
face; there are therefore one favourable and five unfavourable
cases, all equally likely. The required probability is there
fore .
2. What is the probability of throwing a number greater
than two with an ordinary die? Obviously there are 4 possible
favourable cases out of a total of 6. The probability is there
fore or .
3. A bag contains 5 white, 7 black, and 4 red balls. What
is the probability that 3 balls drawn at random are all white?
We have 16 balls altogether. The total number of ways in
which 3 balls can be drawn is therefore 16 C 3 , and the total
number of ways in which 3 white balls can be drawn is
5 C 3 . Therefore, by definition, the probability is 5 C 3 / 16 C 3 ,
that is, 5 3 .
By a compound event , we mean an event which may be
decomposed into two or more simpler events. Thus, fhe
firing of a gun may be decomposed into pulling the trigger,
the fall of the hammer, the explosion of the cartridge, &c.
In this example, the simple events are not independent,
because, if the trigger is pulled, the other events will, under
proper conditions, necessarily follow, and their probabilities
are therefore the same as that of the first event. Events
are independent when the happening of the one does not
render the other either more or less probable than before.
Thus the death of a person is neither more nor less probable
because the planet Mars happens to be visible. When the
component events are independent, a simple rule can be
given for calculating the probability of the compound event,
thus: Multiply together the fractions expressing the proba
bilities of the independent component events.
If, for instance, A occur once in 6 times, its probability
is ^, or 1 for and 5 against; if B occur once in 10 times, its
probability is y^, or 1 for and 9 against. The probability,
or relative frequency in the long run, of the concurrence of
the two is 0 ( y, that is, 1 for and 59 against.
The justification of the rule may be shown thus. If
548 CRAFTSMANSHIP IN MATHEMATICS
two dice are thrown, the side which the one shows upper
most has nothing to do with the side which the other shows
uppermost; but each die has 6 sides, each of which may fall
uppermost, and each of these may with equal possibility
coincide with any one of the 6 sides of the other; there are
thus 36 possible cases, and the probability of each single
one of them is $$ ( = J X ^).
We may add one or two more problems.
1 . What is the probability of throwing an ace in the first
only of two successive throws of a single die? Here we
require a compound even to happen, namely, at the first
throw the ace is to appear, at the second throw the ace is
not to appear. The probability of the first simple event is
, and of the second . Hence the required probability is
A (= * x D
2. A party of 23 persons take their seats at a round table.
Show that it is 10 to 1 against two specified individuals
sitting next to each other. The probability that a given
person A is on one side of a given person B is  2  2 ; the proba
bility that A is on the other side of B is also ^ ; hence, the
probability of A being next to B is  2 5 2 = ^. Thus the odds
are 10 to 1 against A and B sitting together.
3. Find the probability of throwing 8 with two dice.
With two dice, 8 can be made up of 2 and 6, 3 and 5, 4 and 4,
5 and 3, and 6 and 2, that is 5 ways. The total number of
ways is 3G. The probability is therefore  6 , and the odds
31 to 5 against.
4. A pack of 52 cards consisting of 4 suits is shuffled and
dealt out to 4 players. What is the chance that the whole
of a particular suit falls to a particular player?
(3w!) (n\) 1
Chance = v ~~ = approximately,
i.e. 1 in something less than a billion
The Laws of Probability rest upon the fundamental
principles of reasoning, and cannot be really negatived by
any possible experience It might happen that a person should
STATISTICS
549
always throw a coin head uppermost, and appear incapable
of getting tail by chance. The theory would not be falsified
because it contemplates the possibility of the most extreme
runs of luck. But the probability of the occurrence of extreme
runs of luck is excessively slight. Whenever we make any
extensive series of trials, as in throwing a die or coin, the
probability is great that the results will agree pretty nearly
with the predictions yielded by theory. Precise agreement
must not, of course, be expected, for that, as the theory
shows, is highly improbable. Buffon caused a child to throw
a coin many times in succession, and he obtained 1992 tails
and 2048 heads. The same experiment performed by a
pupil of De Morgan's resulted in 2044 tails to 2048 heads.
In both cases the coincidence with theory is as close as could
be expected. Jevons himself made an extensive series of
experiments. He took 10 coins, and made 2048 throws in
two sets of 1024 throws each. Obviously, the probability
of obtaining 10, 9, 8, 7, &c., heads is proportional to the
number of combinations of 10, 9, 8, 7, &c., things chosen
from 10 things. The results may therefore be thus con
veniently tabulated:
Character of Throw.
Theoretical
Numbers.
First
Series.
Second
Series.
Average.
Divergence.
10 Heads, Tails
10 C = 1
3
1
2
+ 1
9 1
10 C! = 10
12
23
17i
+ 71
8 2
10 C 2  45
57
73
65
+ 20
7 3
10 C 3 = 120
129
123
126
+ 6
6 4
10 C 4 = 210
181
190
186J
241
5 5
10 C 5 = 252
257
232
2441
 71
4 6
10 C 6 = 210
201
197
199
11
3 7
10 C 7 = 120
111
119
115
 6
2 8
10 C 8 = 45
52
50
51
+ 6
1 9
10 C 9 = 10
21
16
18
+ 8
10
10 C 10 = 1
1
i
 1
1024
1024
1024
1024
550
CRAFTSMANSHIP IN MATHEMATICS
The present writer repeated the same series of experi
ments, with the following results:
Character of Throw.
Theoretical
Numbers.
First
Series.
Second
Series.
Average.
Divergence.
10 Heads, Tails
10 C  1
4
2
+ 1
9 1 ,
10 d = 10
20
6
13
+ 3
8 2 ,
10 C 2 = 45
40
40
40
5
7 3 ,
10 C 3 = 120
83
150
116i
 3
6 4 ,
10 C 4  210
224
222
223
+ 13
5 5 ,
10 C 5 = 252
250
209
229J
22J
4 6 ,
10 C = 210
242
222
232
+ 22
3 7 ,
10 C 7  120
115
107
111
 9
2 8 ,
>C H  45
28
60
44
 1
1 ,
10 C y  10
14
6
10
10 ,
lo c iy  i
4
2
3
+ 2
1024
1024
1024
1024
The whole number of single throws of coins amounted to
2048 X 10, or 20,480 in all, one half of which, or 10,240,
should theoretically give heads. The total number of heads
obtained by Jevons was 10,352 (5130 in the first series, and
5222 in the second). The number obtained by the present
writer was 10,234 (5098 in the first series, and 5136 in the
second). The coincidence with theory is in each case fairly
close.
Boys should be encouraged to repeat on a small scale a
few experiments of this kind. Their interest is kindled
when they find that a practical result closely approximates a
theoretical estimate.
Correlation
Suppose that a group of measurements give us data about
two variables, say (1) the weight, (2) the stature, of a number
of men. Then we may not only ask questions with regard
to the variation of weight, and questions with regard to the
STATISTICS 551
variation of stature, but we may also raise the further question
of the connexion between the two. A boy who is taller than
another is not necessarily heavier, and yet there is un
doubtedly some connexion between height and weight. This
question of correlation in statistical theory is becoming one
of rapidly increasing importance.
The existence of the connexion itself may, of course, be
in question. Is a boy who is good at sports likely, in the
long run, to be a duffer in the classroom? Some very mathe
matical pupils are, and some are not, musical. Some very
musical pupils are, and some are not, mathematical. Is it
possible to discover a definite measurement of the degree
of connexion between two things whenever the things them
selves are capable of trustworthy estimation?
One of the simplest methods of measuring correlation is
Professor C. Spearman's footrule method; it is easily
mastered in five minutes. Whatever work of this kind may
be attempted with schoolboys, not only the Spearman co
efficient, but the BravaisPearson coefficient, should be
familiar to all teachers.
From the teaching point of view, Spearman's method
possesses the advantage that original material for illustrating
its use is always available in schools. Investigations of the
correlation between the performance of a class in different
subjects, in the same subject in different terms, in different
examinations in the same subject, in school performances
which are not all academic subjects, all these would give
valuable information to the teacher. The use of the corre
lation coefficient as the measure of the " reliability " of an
examination test is of special importance.
The teacher should consult the works of Professor Spear
man, Professor Thorndike, Professor Karl Pearson, Dr. W.
Brown, Udny Yule, and A. L. Bowley.
Statistics has become such a big subject that teachers
may decide against its introduction into schools. But at
the very least boys should be warned of the seriously faulty
552 CRAFTSMANSHIP IN MATHEMATICS
inferences drawn from statistics by the imperfectlytrained
student of economics. University degrees in Economics may
now be obtained by students with only a superficial know
ledge of mathematics. Need we therefore feel surprised at
the absurd economic opinions now often expressed by some
of our younger politicians? One of the commonest political
fallacies is to impose a correlation on two utterly unrelated
graphs, perhaps those concerning (i) foreign trade and (ii) the
marriage rate, on the sole ground that the graphs show some
what similar variations.
Mathematics teachers should warn their pupils that opinions
based on statistics cannot be more than probably true; the
degree of probability may be very great, but there can be no
absolute certainty.
Statistics is beginning to occupy an important place in
theoretical physics. Dirac says: " When an observation is
made on any atomic system ... in a given state, the result
will not in general be determinate, i.e. if the experiment is
repeated several times under identical conditions, several
different results may be obtained. If the experiment is re
peated a large number of times, it will be found that each
particular result will be obtained a definite fraction of the
total number of times, so that we can say there is a definite
probability of its being obtained any time the experiment is
performed. This probability the theory enables us to cal
culate. In special cases, the probability may be unity, and the
result of the experiment is then quite determinate." Instead
of the accuracy and precision which until a short time ago we
have always ascribed to nature, we seem to have nothing but
uncertainty and randomness. Nature seems to know nothing
whatever of simple mathematics. Virtually the presentday
physicist seems to be immersed in the study of the statistics
of electron " jumps ". We can foretell what will happen in the
long run when we throw up coins, and apparently we can quite
definitely forecast what will happen in the long run when we
experiment with vast crowds of atoms and electrons. The
laws of averages and of probability are entering more and more
STATISTICS 553
into the physics of smallscale things. The 2000yearold ques
tion of causation (determinism) presents itself anew.
CHAPTER XXXVII
Sixth Form Work
The Normal Programme for Specialists
The work done by Sixth Form specialists is almost always
work in preparation either for University Scholarships or
for the Higher Certificate. It has become stereotyped in
scope, and much of it has been described as " deadly dull ".
Inasmuch, however, as the University Authorities seem to
require sent up to them boys who have been "well grounded ",
boys who are proficient in the use of those mathematical
weapons which will make attack on the University Course
immediate and effective, mathematical teachers appear to
have no option but to make their boys face the necessary
drudgery. If boys are actually going on to the University,
perhaps that does not much matter. But if they are not,
it is pretty safe to say that their mathematical interest will,
as a rule, cease as soon as they leave school.
In 1904, a Committee of the Mathematical Association,
consisting of 34 of the leading mathematical masters in the
country, reported on " Advanced School Mathematics ".
The committee took into account the different classes of
boys who study advanced mathematics in schools, e.g.
candidates for army examinations, science students, engineer
ing students, and boys who intend to read mathematics at
the University, and they framed a course of instruction which,
they hoped, would prove suitable for all. The following is a
summary:
554 CRAFTSMANSHIP IN MATHEMATICS
1. Algebra: partial fractions, elementary manipulation with
complex numbers and geometrical applications thereof, the
theory of equations so far as it treats of the numerical solution
of equations, the notation and easy properties of determinants,
the simpler tests of convergency, and the binomial, expo
nential, and logarithmic series; but excluding the theory of
numbers, probability, continued fractions, and advanced
theorems on inequalities, on indeterminate equations, and on
summation of series.
2. Differential and Integral Calculus: introduction, and
a free use of the calculus in subsequent work.
3. Trigonometry: graphical illustrations of De Moivre's
theorem, simple work in trigonometrical series and factors.
4. Conic Sections: a treatment of the elementary parts
of the subject, in which either the geometrical or the ana
lytical method is used, that method being used in each
particular case which is most suitable for the problem under
discussion.
5. Solid Geometry: the elementary geometry of the plane,
cone, cylinder, sphere, and regular solids, including practical
solid geometry.
6. Dynamics: an introduction to the dynamics of rotation
(in two dimensions), viz. the motion of a rigid body round
a fixed axis with uniformly accelerated angular velocity,
together with other simple cases of the motions of rigid
bodies.
The Committee urged the importance of " a more intimate
union between the teaching of mathematics and science,
whereby theoretical and practical work may be brought into
relation with one another ".
In 1907, the committee outlined a special schedule of
work suitable for boys preparing for Oxford and Cambridge
scholarships.
1. Pure Geometry. Geometry of straight lines, circles,
and conies; inversion, crossratios, involution, homographic
ranges, projection, reciprocation and principles of duality,
SIXTH FORM WORK 555
elementary solid geometry including plans and elevations of
simple solids.
2. Analytical Geometry. Straight lines and curves of
the second degree; tangential coordinates. Excluding
homogeneous coordinates, invariants, and analytical solid
geometry.
3. Algebra, including elementary theory of equations.
Excluding recurring continued fractions, harder tests of
convergence, theory of numbers, and probability.
4. Geometrical Trigonometry. Excluding spherical trigo
nometry.
5. Analytical Trigonometry. Properties of circular, hyper
bolic, exponential and logarithmic functions. Excluding
the proofs of the infinite products for sine and cosine, and of
the series of partial fractions for the other trigonometrical
ratios.
6. Calculus. Total and partial differentiation; Taylor's
and Maclaurin's theorems; elementary integration; simple
applications to plane curves (especially to such as are of
intrinsic importance), to maxima and minima, to areas and
volumes, and to dynamics; curvetracing, not as a rule to scale.
Excluding the theory (but not the use) of differential equations.
7. Dynamics. Elementary statics, including simple gra
phical statics; elementary kinematics and kinetics, including
motion of a rigid body about a fixed axis, and motion of
cylinders and spheres in cases where the centre of gravity
describes a straight line. Excluding hydrostatics and hydro
dynamics.
In a suggestive article which appeared in the Mathematical
Gazette for January, 1923, Mr. F. G. Hall asks for a much
greater unification of subjects in Sixth Form work. He
points out that the separate consideration of the different
subjects involves a great waste of time. He says that, for
instance, ratio and proportion are treated algebraically,
geometrically, and trigonometrically; logarithms occur in
every type of textbook; variation is considered from the
556 CRAFTSMANSHIP IN MATHEMATICS
point of view of formal algebra, algebraic graphs, trigo
nometrical graphs, and the calculus. He then outlines a
revised scheme, under seven headings:
1. Ratio in algebra, geometry, and trigonometry.
2. The interrelation of trigonometry and geometry.
3. Variation; the general study of functionality.
4. The elements of the differential calculus and its application to
algebra, geometry, and trigonometry.
5. Logarithms and their use in arithmetic, algebra, and trigono
metry.
6. The elements of the integral calculus and its application to
mensuration.
7. Elementary Analysis: (a) the important expansions of algebra
the binomial, exponential, and logarithmic theorems; (p) further
trigonometry, with additional work on De Moivre's theorem and the
expansions to which it leads; (y) easy treatment of the following:
Rolle's theorem and the First MeanValue Theorem; the different
substitutions employed to effect the important types of integration;
successive differentiation; elementary differential equations. The
underlying ideas of this section are (i) the analytical study of " form "
in pure mathematics, and (ii) the development of manipulative power
to enable the study of Higher Mathematics to be undertaken when
the pupil proceeds to a University course.
The whole article is worth reading; it makes fruitful sugges
tions for economy of effort and for the linking up of different
subjects.
Professor Nunn, discussing studies of the kind usually
found in textbooks on " higher algebra ", urges that such
studies do not offer the most suitable course of instruction
for the general body of Sixth Form pupils. " For the student
who is to be a teacher or engineer, or to engage in higher
industrial or administrative work, as well as for the student
who is continuing his mathematical studies as part of a
general education, the best course would seem to be one
which sets in clear relief the central aims and most vital
notions of the main branches of mathematics, supplements
exposition with sufficient practical exercises to give the
SIXTH FORM WORK 557
student a real training and the sense of mastery that comes
with training, and, in particular, illustrates vividly the essential
part which mathematics plays in so many departments of
modern life and activity. "
Excellent advice. Would that it were more generally
followed!
Simpler Fare for the Non Specialists
It will be observed in the last paragraph that Professor
Nunn was speaking of the " general body " of Sixth Form
pupils, whereas both the Committee of the Mathematical
Association and Mr. F. G. Hall had in view the small
section who intend to specialize in mathematics. Now it
is a fact, and a very regrettable one, that Sixth Form boys
who are not specializing in mathematics and science, or at
least the great majority of them, do no mathematics at all.
It is for these boys that I wish to enter a plea. I do not ask
for a supersession of the present type of specialist Sixth
Form work; in practice such supersession is not possible.
The Universities know what they want and schools have no
option but to prepare the boys accordingly. But I do ask
for the provision, for all sections alike of the Sixth Form, say
for two periods a week (assuredly a modest allowance), for
a course of mathematical instruction that shall be at once
less bookish, less academic, more informal, more interesting,
in short, mathematics of the byways rather than the sterner
stuff so much beloved by the successful mathematician.
I want all boys to leave school with a cultivated interest in
mathematics. Some people are of opinion that this may be
effected by making the proposed general Sixth Form course
wholly recreational, but I would make the work more exact
ing than that. I believe it might continue to be as exacting
as the work already done in the Fifth. Nevertheless, I would
willingly sacrifice much of the formality of the subject in
order to ensure a permanence of interest. The formal
training I would willingly subordinate to interest and
558 CRAFTSMANSHIP IN MATHEMATICS
knowledge, especially knowledge of unsuspected points of
contact between mathematics and nature.
With this purpose in view I throw out a few hints in
seven short chapters:
1. Harmonic Motion.
2. The Polyhedra.
3. Mathematics in Biology.
4. Proportion and Symmetry in Art.
5. Numbers; their unexpected Significance.
6. Time and the Calendar.
7. Mathematical Recreations.
CHAPTER XXXVIII
Harmonic Motion
The Ordinary Book work of S.H.M.
The work commonly done in connexion with Simple
Harmonic Motion (which has already been touched upon
in Chapter XXXI) usually begins
in this way:
Let P travel with uniform speed
round the circumference of a circle.
If Q is the foot of the perpen
dicular on the diameter AB, the
motion of Q is an oscillation be
tween A and B. Q is the projec
g. 289 tion of P on AB, and the motion
of Q is Simple Harmonic Motion.
Thus we may define S.H.M. as the projection of a uniform
circular motion on to a diameter of the circle.
Then usually follows a series of blackboard demonstrations:
HARMONIC MOTION 559
Let the amplitude of the oscillation, i.e. the radius of
the circle, be a.
Let ft be the square of the angular velocity of the point P
about the centre O.
(i) Since the angular velocity of OP =
/. the linear velocity of P a\//x
The acceleration of Q /z.OQ towards O;
acceleration of Q
or, . =
_ acceleration of Q
displacement
(ii) The period of oscillation .
(iii) The velocity of the point Q (in any position)
"
(iv) The period of a small oscillation of a simple pendu
lum = 2?r A/  .
of
(v) The length of a seconds pendulum = / %>
This is all sound enough, but to many boys it is just an
affair of algebra, with only the vaguest relation to practical
life. The boys probably forget their practical exercises on
loci in geometry when from models of linkages they learnt
how a toandfro motion may be converted into a circular
motion, or, if they remember, they do not associate the new
work with the old. They are probably not made to realize
that S.H.M. forms the basis of the investigation of most
oscillatory movements that occur in nature, such as the
small oscillation of a simple pendulum, the vibration of
strings and other bodies emitting musical notes, the light
vibrations in the aether, the vibrations producing Fletcher's
trolleywave; and so on. They have been studying S.H.M.
in their mechanics and physics without knowing it, and
now they are studying S.H.M. as something having only the
vaguest relation to their mechanics and physics.
560 CRAFTSMANSHIP IN MATHEMATICS
What is the use of playing about with the mere algebraic
formula of simple harmonic motion before the motion has
been studied and its full significance grasped in the related
practical work in mechanics and physics? The laboratory,
not the classroom, is the place for teaching S.H.M.
S.H.M. Experiments
Elaborate experiments are not necessary, Here are a
few simple ones.
1. Let the boy stand facing a white wall, or a lantern
screen, and at the back of the room let a strong light be
placed at about the level of his head. His shadow will be
cast on the wall or screen. Now let him whirl, in a horizontal
plane, round his head a small heavy body attached to the
end of a string about a yard long. After a little practice he
can maintain his hand in almost the same position, and
keep on whirling at a fairly constant velocity. As the small
heavy body moves round his head, its shadow moves to
and fro on the screen. There is a true projection of the
circular motion, the shadow showing a true S.H.M. (A
conical pendulum may also be used. Let the heavy body
first hang vertically, then cause it to swing in such a way
that the pendulum describes a conical surface. Note that
the effective length of this pendulum is the vertical depth
of the weight below the point of support.)
2. Let the boy continue to whirl the stringpendulum
around his head; then set up an ordinary pendulum between
him and the wall, and let it vibrate. The latter may be allowed
to swing with rather more than " small " vibrations; these
will not, it is true, be in strict S.H.M., but for a rough experi
ment that will not much matter. Now adjust the length
of the string in the first experiment so that the toandfro
movement of its shadow is about equal in length to the
swing of the suspended pendulum, and so adjust the time of
whirling that there is one complete revolution to a complete
toandfro movement of the suspended pendulum. With
HARMONIC MOTION
561
a little practice (it needs practice), the shadow of the heavy
weight may be made to follow exactly the movement of the
pendulum bob (except of course that the latter does not
move quite in a horizontal plane). The experiment convinces
the observer that the movement of the shadow is syn
chronous at all points with the movements of the pendulum
bob.
3. Mount on a board a good sine curve as produced by a
Fletcher's trolley running with uniform speed. Cut a slit
in a piece of cardboard and arrange the cardboard so that
Fig. 290
its midline AB always coincides with the axis XY of the
curve. Move the cardboard to and fro with uniform speed x.
The bit of curve showing through the slit seems to move up
and down with S.H.M. A permanent model is worth
making in the school workshop. A groove may be run along
the board, coinciding with the XY axis, and at the back of
the cardboard (a thin piece of wood is better) two bits of
wood are fixed at AB, one on each side of the slit, to slide
along the groove. A better model may be devised by mount
ing the curve on a large wooden cylinder which is made to
revolve, the cardboard with the groove being fixed medially
in front of it. The resemblance of the bit of line moving
up and down the groove to a pendulum movement is very
striking.
4. Revise the laboratory experiments on the pendulum,
and verify that the time of vibration of a pendulum varies
as the square root of the length.
Lead the boys to see clearly that Fletcher's vibrating
lath is really a pendulum, and the trolley simply a device
for recording its vibrations pictorially; and that the model
(291) 37
562
CRAFTSMANSHIP IN MATHEMATICS
in (3) above is, in its turn, another device for showing the
lath vibrations more slowly, so that we can examine the way
in which the lath really did move.
The term " harmonic " is justified because when the
vibrations of the lath follow one another with sufficient
rapidity, a definite musical note is heard which rises higher
in pitch with increasing rapidity of the vibrations. So with
vibrating things generally.
Compound Harmonic Motion
This subject is important in connexion with the study
of sound. Lissajous curves are produced by compounding
harmonic motions. The " compound pendulum " produces
many of them in a simple and
sufficiently effective manner.
Here the mathematical master
will be able to get help from
the physics master.
A pendulum dropping sand
affords the simplest means of
illustrating the combined mo
tions. A funnel F containing
sand is suspended by two
strings, passing through two
small holes in a cork C, from
two hooks A and B in the beam
of the frame MN, and swings
in S.H.M. at right angles to the
frame; the sand dropped from
the funnel makes a line on the
paper below, also at right angles to the frame. If the
paper is now steadily drawn across the baseboard with
correct time adjustment, in a direction parallel to the frame,
the sand will trace out a sine curve. Now draw the funnel
aside so that it swings in the plane of the frame and the
sand makes a line in that plane; the swing is again in S.H.M.;
Fig. 291
HARMONIC MOTION 563
slide the paper across the board, perpendicularly to the
plane of the frame, and another sine curve results, but at
right angles to the former. Finally, draw the funnel aside
in a direction neither parallel nor perpendicular to AB (it
might have been fastened back by a piece of cotton attached
to the upright Z, fixed in a 45 position, and suddenly
released by cutting or firing the cotton). The new motion
combines the characteristics of the two former S.H.M.s,
and the motion is said to be compound H.M. The precise
form of the movement will depend (1) on the ratio of the
two S.H.M.s, that is, the ratio of the lengths of the two really
separate pendulums DE and CE, D and C being the re
spective points of support; and (2) the direction in which
and the distance to which the funnel is drawn back. We
may modify the first of these factors by sliding the cork C
up and down the string; the length of the shorter pendulum
will always be equal to CE. Of course the paper will not
now be moved at all.
The pupils must understand clearly that if the pendulum
swings at right angles to the frame MABN, its effective
length is DE; that if it swings parallel to the frame its effective
length is CE; that therefore by altering the position of C
we may make the ratio of the two lengths any value we
please.
Let the ratio be 1:4, e.g. let CE be 9" and DE 36".
Set in motion by releasing from Z. The bob (the funnel)
of the two pendulums cannot move in two
directions at the same time and it therefore
makes a compromise and follows a path
compounded of the two directions. It
traces over and over again fig. 292. Since
the times of vibration vary as the square
roots of the lengths of the pendulum, these ^ Fig. 292
times are as \/l : y/4, i.e. 1 : 2. Thus the
short pendulum CE swings twice while the larger one DE
swings once.
If we wish the times of vibrations to be 2 : 3, that is if
564:
CRAFTSMANSHIP IN MATHEMATICS
we wish the larger pendulum to swing twice while the shorter
swings three times), the ratio of the pendulum lengths must
be 2 2 : 3 2 or 4 : 9, i.e. the shorter pendulum
must be 4/9 of the larger one. If the larger
one DE is 36", the shorter one CE must
be 16", and the cork C must be adjusted
accordingly. The sand curve now traced,
over and over again, is shown in fig. 293.
Let the pupils try other simple ratios.
Here is a short table of musical intervals
formed by two notes which are produced
by numbers of vibrations bearing to each
other the same ratios as those given in the first column.
Fig. 293
Frequency
Ratio.
Pendulum
Length Ratio.
Corresponding
Musical Interval.
1 2
2 3
3 4
4 1
9 4
16 9
Octave (fig. 292)
Fifth (fig. 293)
Fourth
4 5
5 6
25 16
36 25
Major Third
Minor Third
3 5
25 9
Sixth
The new and more elaborate figures will interest the pupils,
who must, however, realize that in all cases the movement
actually executed is the resultant of two S.H.M.s perpendicular
to each other.
In practice it is virtually impossible to set the two pen
dulums at the exact ratios given. The simple curves are
therefore not maintained, but they open out and close again
in a curious but regular movement. Here are examples of
the changes seen in figs. 292 and 293, representing the octave
and the fifth. But within a short time the curves are lost in
the gradually spreading sand. The wellknown successive
phases of Lissajous figures, due to the tuningforks not
being in exact unison, are identical with the figures here
shown, and, for all practical purposes, are produced in the
HARMONIC MOTION
565
Fig. 294
same way. The boys should see them produced in the physical
laboratory.
Obviously a better device than the swinging sand funnel
is wanted for producing
the figures. Pupils often
suggest the substitution
of a pencil or a pen for
the sand, but of course
this is not possible,
inasmuch as the funnel
does not move in a hori
zontal plane.
This difficulty has
been overcome by devis
ing an entirely new type
of compound pendu
lum. Two pendulums
are hung from a small
wooden table supported
on three legs, the pendu
lum rods passing through Figt 295
large holes in the table
top so that they can swing without touching it. To the top
of one, a small table with paper is fixed so that it moves to
566 CRAFTSMANSHIP IN MATHEMATICS
and fro while the pendulum swings. The top of the other
carries a long rod which in turn carries a glass pen. Each
pendulum can swing in one direction only, like a clock
pendulum. Each pendulum swinging alone would simply
describe a straight line of ink, again and again, until the
motion died away. If the two pendulums are made to swing,
not together, but in succession, there will be two straight
lines at right angles to each other. But the two may be made
to swing together, each to record whatever impulse may
first be given to it, and so we get figures of the same type
as the sand figures.
As the pen moves in a vertical plane, the surface of the
receiving table top of the other pendulum is given the neces
sary slight cylindrical curvature, in order that the pen may
always be just in contact with it.
We may vary as we please the impulses given to the
pendulums. We may start them either at the same time, or
one rather later than the other, say when the first has covered
some definite fraction (J, , or f) of its path. We may vary
the length of either pendulum by raising or lowering the
weights (bobs); the effective length of a pendulum is from
the point of suspension to the centre of the bob.
Thus the figures produced may be varied greatly.
If the two pendulums are exactly equal, and if we start
them swinging at exactly the same instant, we shall obtain
one of these figures:
Fig. 296
These are really only different phases of the same figure, and
they form the simplest group of Lissajous figures: all are
ellipses, the straight line and the circle being merely par
ticular cases. If the two are started at the same pace and
at the same moment, the figure is a straight line; if the one
HARMONIC MOTION
567
begins its movement when the other has already completed
half its path, the figure is a circle; if the difference is greater
or less than half a path, then an ellipse.
But although the pendulum continues to move, the pen
does not continue to mark out exactly the same line, circle,
or ellipse. We have to take friction into account. When
the pen comes round to the point where, say, the circle
began, it just misses it, and the second circle begins inside
the first: really we get a spiral, ending at the centre of the
paper when the pendulum comes to rest. By varying the
friction, varying the ratio of the pendulums, and varying
the impulses, we may obtain figures of extraordinary beauty.
Many of these, exquisitely reproduced in colour, may be seen
in Newton's Harmonic Vibrations and Vibration Figures. The
more elaborate pendulums which have been designed for
producing such figures (Benham's triple pendulum, for in
stance) are worth a careful examination for their mechanical
ingenuity alone.
The mechanical difficulties associated with the compound
pendulum (the harmonograph) are many, and only the
exceptionally patient teacher is advised to purchase one.
The feeding of the glass pen with a suitable ink, and the
adjusting of the pen, are particularly teasing operations.
In two or three cases
I have known the beauti
ful figures make a strong
appeal to pupils who had
previously professed their
dislike of mathematics.
Various other instru
ments have been designed
for producing curved de
signs mechanically. Fig.
297 shows two designs pro
duced by the Epicycloidal
Cutting Frame, but, gene
rally speaking, the mathe Fig. 297(0
568
CRAFTSMANSHIP IN MATHEMATICS
Fig. 297 (")
matics of such designs, though not advanced, is too tedious and
elaborate to render its introduction into schools worth while.
Geometrical Construction of the Compound Curves
Let the pupils draw a few of the simpler figures geo
metrically.
They may refer again to
the pendulum whirling round
the head. Let a revolution
take two seconds, so that the
pendulum rotates through y^
of the circle in g of a second.
To an observer watching the
shadow on the screen, the
bob appears to travel from
1 to 2 in the straight line,
while it really travels from 1
to 2 in the circle. It appears
to travel with greatest velocity
at 1 and 9 in the straight
Fig . 29 8 line, and to be momentarily
\
1514 15
12 II
16
10
I
9
45
6
HARMONIC MOTION
569
at rest at 13 and 5; also to travel from A to B in 1 second,
though in that time it has really travelled halfway round
the circle. A comparison of the unequal distances covered
in equal times, in AB, serves to show the varying velocity
of a simple pendulum.
Remember that the phase of any point in S.H.M. is the
fraction of a period that has elapsed since the point last
passed through the position of maximum displacement.
Fig. 299
Here is a figure (fig. 299: an ellipse) produced by two
S.H.M.s of the same period but differing in phase by \ of
a period. Semicircles suffice for obtaining the necessary
projectors. The points for the ellipse are determined by the
intersection of the two sets of projectors.
Projectors from a and 3 (^ of a phase apart) produce the
point m.
Projectors from b and 2 (^ of a phase apart, yg in each
circle) produce the point n. And so on. The curve follows
the route marked out by the diagonals of successive parallelo
grams.
570
CRAFTSMANSHIP IN MATHEMATICS
Boys who are poor draughtsmen sometimes make a hash
of more complex figures, generally, however, because they have
not grasped the main principle which is the same for all cases.
Here is another figure (fig. 300) showing the compounded
motions of two pendulums whose lengths are as 16 : 25 and
whose vibration ratio is therefore 4:5. Divide the semicircle
EEC into 8 equal parts and the semicircle ADB into 10.
Fig. 300
Project on to the respective diameters, and continue across
the square so that the projectors intersect as shown. The
phases are intended to be the same, so that the two pendulums
may be supposed to start together at B. The pendulum swing
ing towards A would reach M when the pendulum swinging
towards C reached N i.e. if the two were swinging in
dependently; but inasmuch as they are moving together
their motions are compounded; neither is able to take its
own path, and the path actually followed is BR.
Observe that all the separate bits of line, horizontal and
vertical, in the square, represent equal intervals of time.
HARMONIC MOTION 571
Hence every point of intersection through which the curve
passes is determined by compounding two distances re
presenting equal intervals of time. But after the point Q,
we have, of course, to begin to allow for a reversal of direction.
The curve simply follows the route marked out by the diagonals
of successive parallelograms.
Mathematically, the subject is hardly worth following
very far, unless the pupils are seriously studying the subject
of sound in physics. But there are a few exercises in Nunn's
Algebra that all Sixth Form pupils might profitably work
through.
Spirals
This subject of spirals, though outside the scope of the
chapter, may be conveniently mentioned here.
A little work on spirals is worth doing, if only to emphasize
the fact that there is a practical side to the notion that angles
may have any magnitude. For instance, in Archimedes 9
spiral, r = ad\ in the Logarithmic spiral, r = a B \ in the
Hyperbolic spiral, r9 a in the Lituus, r*9 a. A little
work on roulettes should also be done cycloids, epicycloids,
hypocycloids, but not trochoids. These are all best treated,
for the most part, geometrically. Such peculiarities exhibited
by curves as asymptotes, nodes, cusps, points of inflexion,
&c., should be explained.
All these curves are full of interest; dwell on that side
of them.
Spirals afford a good start for the study of vectors. In
the Archimedean spiral, equal amounts of increase in the
vectorial angle and radius vector accompany one another,
i.e. if one is in A.P., so is the other. In the logarithmic spiral
if the vectorial angles form an A.P., the corresponding radii
form a G.P.
Books to consult:
1. Any standard work on Sound
2. Harmonic Vibrations, Martin.
572
CRAFTSMANSHIP IN MATHEMATICS
CHAPTER XXXIX
The Polyhedra
Euclid, Book XIII
It is to be regretted that the displacement of Euclid
has led to the abandonment of the greater part of the substance
of his 13th book. It is true that the five regular solids are
referred to in most modern textbooks of geometry, sketches
of them given, " nets " for their development suggested,
and perhaps Cauchy's proof of Euler's relation E+2 = F+V
worked out. But assuredly rather more than this ought to
be done. The intimate relations of the five solids to each
other, and the almost equally close relations between them
and their many cousins, are so remarkable, and at the same
time are so simple, that all boys ought to know something
about them. The work forms a fitting completion to the
course of solid geometry.
A regular polyhedron is one all of whose faces are equal
and regular polygons, and all of whose vertices are exactly
alike and lie on the circumscribed sphere. There are only
5, and the proof that there cannot be more is easy and should
be provided. They are:
Name.
Number and
Nature of Faces.
Sum of Plane /.s
at Vertex.
Remarks.
Tetrahedron
4 equilateral As
60 X 3 = 180
} Found as
Cube
6 squares
90 x 3 = 270
> natural
Octahedron
8 equilateral As
60 X 4 240
] crystals
Dodecahedron
Icosahedron
12 pentagons
20 equilateral As
108 X 3  324
60 X 5 = 300
\ Artificial
Closely related to the 3 natural polyhedra are 2 other
natural solids, the rhombic dodecahedron (faces = 12 rhombuses),
and the trapezohedron (faces = 24 kites). And closely related
THE POLYHEDRA
573
to the 2 artificial polyhedra are 2 other artificial solids, the
triacontahedron (faces 30 rhombuses), and the hexacon
tahedron (faces = 60 kites). In all 9 cases, E + 2 = F + V.
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
Fig. 301
574
CRAFTSMANSHIP IN MATHEMATICS
On the previous page are sketches of the 5 polyhedra and
nets for their construction. Third Form boys make them
up readily enough (for practical hints, see Lower and Middle
Form Geometry, pp. 2023).
In my own teaching days, we taught the interrelations of
all 9 solids, but in these days of high pressure, one has to
be content with a few main facts about the 5 regular solids
alone. For those who wish to construct models of the
A other four, the
following di
mensions of the
respective rhom
buses and kites
may be useful
Fig. 302
* J"
Rhombic dodecahedron,  =  .
V ' +V. A MP 2
Inacontahedron, = .
(fig. 3 2 ) :
T . U A MP
Tnacontahedron, _
,, AC 3V3 AE 1
Trapezohedron, _ = ^^ = .
  AC V25 + 5V5 AE
Hexacontahedron, _ =  ;
The key propositions to Euclid XIII are 1318. The
first five of these show how to construct the polyhedra, how
to inscribe each in a sphere, and how to express the length
of an edge in terms of the radius of the sphere. If E be the
edge, and R be the radius of the sphere,
Proposition 13, tetrahedron, E =  \/6 . R.
14, octahedron, E = V 2 R 
15, cube, E = fi/3 . R. _
16, icosahedron, E = Vl<K/5(<v/6l) R 
17, dodecahedron, E = ^ ( V5  1)R.
THE POLYHEDRA
575
Encourage the pupils to seek for interrelations amongst
these. For instance, compare Propositions 15 and 17; in
the latter, E = (V 5 1) of E in the former. Hence, if
the edge of the inscribed cube be cut in golden section, the
length of the greater segment is the length of the edge of the
inscribed dodecahedron.
Euclid XIII, 18, is also useful. To set out the edges of
the five regular solids and to compare them with one another
and with the radius of the circumscribing sphere.
Cutting One Polyhedron from Another
We shall refer to polyhedra " contained " within other
polyhedra. The following volume relations should there
fore be noted:
Contained Solid.
Containing Solid.
Octahedron.
Cube.
Tetrahedron.
Octahedron
9:2
27 :2
Cube
6: 1
3:1
Tetrahedron
2: 1
9: 1
The mutual relations of the two artificial solids are less simple.
It may be noted that the ratio of the edges of an icosahedron
and its contained dodecahedron is 6 : 1 j~ y/5.
A group of interesting problems are those of calculating
the angles between edges, between faces, and between edges
and faces, in the polyhedra. They are good problems, but
some are a little tedious. In this connexion, the socalled
15th book of Euclid may be usefully consulted.
Practical demonstrations of polyhedral interrelations always
appeal to boys. Encourage the boys to make their own models
and to discover how easy it is to cut all sorts of regular sections
through them. For instance:
1. A cube may be cut into two equal parts by means
576 CRAFTSMANSHIP IN MATHEMATICS
of a section of the form of a regular hexagon; the section
passes through the midpoints of 6 of the 12 edges of the
cube.
2. A hexagonal section of an octahedron may be made
similarly.
3. A hexagonal section of a dodecahedron, dividing this
solid also into 2 similar halves, may be made by joining any
2 opposite angles of any pentagonal face (i.e. by drawing
any one of the 5 diagonals of any face), and passing a plane
through this join and the centre of the solid. The 6 edges
of the hexagon are formed by 6 similar joins.
If such sections as the last are taken in all possible ways
through a polyhedron, the intersection of the new planes
will give clues to all sorts of interesting variations.
If time can be spared (e.g. after examinations, at the
end of a Term), good preliminary work can be done in a
Third or a Fourth Form in preparation for the more formal
work in the Sixth. We may quote a few sections from Lower
and Middle Form Geometry:
Transformations of Polyhedra
481. Any regular polyhedron may be cut from any other
polyhedron, and the conversion is always a very simple
matter. Suppose you cut a vertex (corner) from a cube,
cutting the three edges back to the same extent; the section
is an equil. A . Suppose you cut a vertex from an icosahedron,
cutting all five edges back to the same extent; the section
is a regular pentagon. Suppose you cut an edge from a cube,
cutting back the faces symmetrically; the section is a rect
angle. This is the sort of thing to be done in the following
experiments. In one case, all the vertices will be cut off;
in another, all the edges; always symmetrically and equally.
The best material to use is best yellow bar soap, from
which you can easily cut cubes. It is better and cleaner
than clay or plasticine. (If you are clever with carpenters'
tools, a finegrained wood is better still.) For cutting soap
THE POLYHEDRA 577
a very thinbladed knife is desirable. Do not attempt to
cut off " chunks "; cut off shavings and do the cutting
gradually.
482. To cut a tetrahedron from a cube. A cube has
eight vertices, a tetrahedron has four. The conversion is
made by cutting away four of the eight vertices; the four
new planes will be the four faces of the tetrahedron.
Draw the diag. AC of the top of the cube, and the opposite
diag. FH in the base of the cube. These are two edges of
the tetrahedron. Mark them by pushing into the soap three
or four small pins in each line. Cut off the vertex D, and
keep paring away until you reach the
plane ACH, one of the faces of the tetra
hedron. Scratch the letters A, C, and
H on this new plane, or the model may
become confusing. Now cut off the ver
tex G until you reach the plane CFH,
a second face of the tetrahedron; now
the vertex B, until you reach the plane Fig. 303
AFC, a third face; lastly, cut off the
vertex, E behind, until you reach the plane AFH, the fourth
face. Now the cutting is completed. Note that the four
vertices cut off are the four not concerned with the two
diagonals first drawn. The two vertices at the end of each
of these diagonals remain, and are the four vertices of the
new tetrahedron.
The figure is a little difficult to follow, but the soap
model itself will keep telling its own story, especially if the
vertices are all lettered as in the figure.
483. To cut an octahedron from a cube. This is done
by cutting off the eight vertices of the cube, the eight new
planes forming the eight faces of the octahedron.
Begin by marking in the six vertices of the octahedron.
They are at " centres " (intersections of the diagonals) of
the six faces of the cube. Show them by small pins thrust
into the soap, K, L, M, N, P (and Q in face ABFE, not
shown). Begin by cutting off the corner C symmetrically,
(B291) 38
578
CRAFTSMANSHIP IN MATHEMATICS
preserving the equil. A all the time, and pare down until
you reach the plane made by the three pinheads K, L,
and M. KLM is one of the eight faces of the octahedron.
Cut off the other seven corners of the cube similarly, and
so obtain the other seven faces. Fig. 304 (ii) shows what
the model would look like if the eight corners were cut
down only to the midpoints of the edges. To make the
octahedron, the cutting has to be continued to the centres
Fig. 304
of the faces, by which time the faces of the cube will have
disappeared.
484. To cut a dodecahedron from a cube. To
construct a pentagon, Euclid used golden section. The faces
of a dodecahedron consist of pentagons, and we shall use
golden section for the necessary construction in this problem.
Golden section cuts a 2" line into two parts, <\/5 1
and 3 \/5 inches long, i.e. 124 and 76 inches, very
nearly. A line 1" long would be cut into parts of 62 and
38 inches, and a 3" line into 217 and 133 inches. We
will cut the dodecahedron from a 3" cube. It is a con
venient size for handling. If you cannot obtain a piece of
soap or other material big enough, you must cut a smaller
cube, but in the same proportion.
The twelve faces of the dodecahedron are the twelve
new planes formed by cutting off the twelve edges of the
cube. Hold a dodecahedron by two opposite edges between
finger and thumb. These two edges may be regarded as
medially placed in the top and bottom faces of the cube.
THE POLYHEDRA
579
They are the middle parts of the medians of the faces of
the cube. The four other edges that occupy medial positions
in the other four faces of the cube can be easily traced as
the model is thus held in the fingers.
The ratio of the length of the edge of the dodecahedron
to the length of the edge of the containing cube is the ratio
of the shorter section to the whole line in golden section.
In a 31" line, the shorter section is 133 in. Call this
Fig. 305
\\ in. Place a line of this length centrally, as shown, in each
of the six faces of the cube. (The three not shown in the
figure they are at the back and underneath correspond
in position to their opposite neighbours.) These six lines
we call " medial " lines.
Fig. 305 (i) shows the construction for each square face.
Since PQ = 31' and RS = 1", PR  SQ = l^V".
The six medial lines of 1J" are six of the thirty edges of
the dodecahedron. The other twentyfour edges will appear
when the twelve new planes have been cut.
Mark the six medial lines by rows of three or four pins
in each, one pin being placed at each end (G and H for
580 CRAFTSMANSHIP IN MATHEMATICS
instance). These can be withdrawn as soon as the cuts nearly
reach them, in order that the cuts may leave sharp edges.
Begin by cutting off the edge AB, cutting down to the
median CD containing MN in the top face, and to the point G
of the medial line GH in the side face. The new plane is shown,
shaded. Obtain the other eleven planes in similar fashion,
every plane extending from one medial to the next, line to
point, as CD to G, the parallels CD and EF guiding the
knife. Fig. 305 (iii) shows, separately, another edge cut
off, and iv shows a third. As one new plane cuts through
another, a part of the latter will disappear. Confusion will
arise unless the points in the model are lettered to corre
spond with the letters in the figure.
It is best to cut off four parallel edges of the cube first
(these four sections will be rectangles), then a second set
of four parallel edges, then the third. The gradual formation
of the dodecahedron will then be more clearly understood.
The pentagons will appear as the last four planes are being
cut.
485. To cut an icosahedron from a cube. We might
proceed as in the last case, first putting into medial position,
in the six faces of the cube, the longer portion of a line
divided in golden section, i.e. 217 in. in a cube of 3 in.
edge. If an icosahedron be held by two opposite edges
between finger and thumb, it is easy to see from the symmetry
of the solid that these two and four other edges occupy
medial positions in the six faces of the cube, exactly as in
the last case. To obtain the twenty faces (equil. As) of the
icosahedron, we have to cut off not only the twelve edges
but also the eight vertices of the cube, the new planes by
their intersections giving the faces. And all the cuts are
determined by the medial lines in the six faces of the cube.
But unless the edges, old and new, are carefully marked,
the cutting is a complicated business. A simpler method
is first to cut a dodecahedron from the cube ( 484). To
cut an icosahedron from a dodecahedron is then easy.
It is done by cutting off the twenty vertices of the
THE POLYHEDRA 581
dodecahedron, continuing the cutting until the twenty new
planes meet to form twenty equil. As. Since each vertex
of the dodecahedron is made up of three equal plane /.s,
an equil. A is produced by cutting off a vertex symmetrically.
Each cut must be continued until the new plane passes
through the centres * of the pentagons. Hence, with
twelve pins, mark the centres of the twelve pentagons; they
show the positions of the twelve vertices of the icosahedron
to be formed. If you begin by first carrying all the cuts
to the midpoints of the edges, a very pretty solid is formed,
consisting of equil. As and small pentagons. But the cuts
must be carried deeper, until they pass through the centres
of the original pentagons.
486. To cut a dodecahedron from an icosahedron.
This case is almost identical with the last. Cut off a vertex
of the icosahedron, symmetrically, and a regular pentagon
results. Cut off all twelve vertices, and twelve pentagons
result. Continue the cutting until each pentagon passes
through the centres* of the AS. Hence, with twenty
pins, begin by marking the centres of the twenty As; they
show the positions of the twenty vertices of the dodecahedron.
If you begin by first carrying all the cuts to the midpoint
of the edges, a pretty solid is formed, consisting of pentagons
and small equil. As. But the cuts must
be carried deeper, until they pass through
the centres of the original equil. As.
487. To cut a cube from a dode
cahedron. Let AB be an edge of a do
decahedron, AC, AD two edges, radiating
from A; BE, BF two radiating from B. If,
in the solid, CD, DE, EF, FC are joined, FI. 300
a square is formed, and this is one of the
faces of the contained cube. It is easy to mark off the other
five faces, in a similar way, and then to cut out the cube.
Reductions of this kind lend themselves to useful geo
* I.e. the intersection of the J_ bisectors of any two sides. In the case of the
A , it coincides with the centroid.
582
CRAFTSMANSHIP IN MATHEMATICS
metrical investigations by Sixth Form boys. Here, for
instance, is an isometrical projection of a cube, showing
all the construction lines for reducing the cube to an icosa
hedron. If the construction lines are based on the 3 initial
lines ST, VW, XY, the angle at H (1382) should be
checked by measurement; and vice versa. The advantage of
the isometric projection is that all lengths parallel to the
edges of the cube are true lengths. It is an excellent
exercise for boys to make cardboard models of the successive
transition stages of reduction, e.g. the edge AF is first
removed down to the plane RabH; then the corresponding
edges through B, C, and D. The face of the model with the
4: edges removed is then Hbcjde. The other 2 groups of 4
edges each are treated similarly. Or, if it be decided to remove
the 8 vertices first, symmetry will decide at once that, e.g.,
the plane cutting off the vertex A must pass through X, T,
md V, and that Af = Ag Ah. The real trouble comes
when some of the edges or vertices have been cut away;
the whole thing then seems to be " lost ". The important
THE POLYHEDRA
683
thing is to keep the 3 initial lines ST, VW, XY in view all
the time. When a model in soap is made, these initial lines
may be marked at the outset by thrusting small pins into the
soap, so that the heads act as guides all through the cutting
process.
The following data are useful when effecting polyhedral
transformations.
Points and Lines in these containing Polyhedra
become in the contained Polyhedra
the following Points and Lines:
( The 4 Cd's of the 4 F's
Tetrahedron [ The 4 F's
1 The midpts. of the 6 E's
( The 4 alt. V's
Cube \ The 6 D's, i in each F
I The 6 C's of the 6 F's
Octahedron. ( The Cd's of the 8 F's
I The Cd's of the 4 alt. F's
Icosahedron. The Cd's of the 20 F's
Dodecahedron. The C. of G. of the 12 F's
the 4 alt. V's of the cube.
4 of the 8 F's of the octahedron.
the 6 V's of the octahedron.
the 4 V's of the tetrahedron.
the 6 E's of the tetrahedron.
the 6 V's of the octahedron.
the 8 V's /of the cube.
the 4 V's of the tetrahedron.
the 20 V's of the dodecahedron.
the 12 V's of the icosahedron.
E = edge; F face; V = vertex; C = centre; Cd = centroid; D = diagonal
A boy who can use the soldering iron may make some
skeleton models in stout brass wire, fairly deep notches
being cut with a file symmetrically at each vertex and else
where as required. Contained models may be exhibited
in position by means of stout threads run continuously from
vertex to vertex, from midpoint to midpoint, and so on.
When the contained " solid " is completed, the thread is
brought back to the first vertex and tied. The different
contained models may be constructed in threads of different
colours, and each is then easily distinguished from its neigh
bours. We showed above how a cube might be cut from
a dodecahedron: evidently 5 such cubes may be so cut,
according to the edge we select to begin with. If all 5 cubes
be threaded in a wire model of the dodecahedron, the re
sultant triacontahedron, with its 30 rhombuses, is effectively
584 CRAFTSMANSHIP IN MATHEMATICS
shown. With care a 3rd polyhedron may be shown within a
2nd, and a 4th inside the 3rd, but girls are generally more
expert than boys in handling and tying the threads.
The " centres " and centroids of the faces of the initial
model may be shown by soldering into position diagonals
or medians of very thin wire.
A useful present to a keen mathematical boy is Bruckner's
Vielecke und Vielflache, Theorie und Geschichte (Teubner).
The many plates contain scores of photographs of beautiful
models based on the polyhedra. The models are easily
made in cardboard or stiff paper (I have known excellent
specimens made by boys during the holidays), and the
accompanying explanations and theoretical matter are well
within the range of a Sixth Form boy. Here are reproduc
tions of four of the photographs.
CHAPTER XL
Mathematics in Biology
General Ignorance of the Subject
There is a remarkable ignorance on the part of the average
person in regard to the numerous matters of mathematical
interest in botany and zoology. And not all mathematicians
have interested themselves in this department of their subject,
even though they may be perfectly familiar with the other
branches of what they sometimes call " applied " mathe
matics.
The following topics are suggested for inclusion in the
general mathematical course for Sixth Form nonspecialists.
For such boys only an elementary treatment will be possible,
though the specialists, if they could spare the time, might
carry the work much further. Each topic, animal locomotion,
Fig. 308. Polyhedral Models
[Facing p. 084
MATHEMATICS IN BIOLOGY 585
for instance, is really a very big subject, much too big for
an exhaustive treatment. But for a boy to leave school entirely
ignorant of the mathematical significance of the facts enu
merated in the following paragraphs is a sad reflection on
the narrowness of the course of mathematical work that
schools commonly provide.
Biological Topics for Consideration
The principles of similitude in biological forms and
structures. Why there is necessarily a limit to the size of
all plants and animals, as well as to all artificial structures.
Stable equilibrium. How Eiffel secured an even distribution
of strength in his tower by adopting the form of the logarith
mic curve.
Mechanical efficiency. The stream lines of a fish and
the lesson to the naval architect; the stream lines of birds
and the lesson to the student of aeronautics. The human
skeleton from the engineer's point of view compression
and tension lines in the construction; ties (ligaments, tendons,
muscles, &c.) and struts. The structure of a few of the
principal bones of the body regarded as engineering units,
and the mechanical distribution of compression and tensile
stress. Compare the skeletal framework of a quadruped with
a bridge supported by two piers: the quadruped is really
an admirably jointed and flexible bridge. Strength and
flexibility in aquatic animals. The remarkable strength of
insects in proportion to their size.
Animal locomotion. Locomotion on land, in air, in
water. The wing regarded as a helix. The relation between
the work which a bird does in moving itself forward and
the linear dimensions of the wings. The minimum necessary
speed in flight.
Rate of growth in the organic world. Rate at different
periods of " life "; its variability and its periodic retarda
tion. The weightlength coefficient. (Growth graphs should
receive special attention.)
586 CRAFTSMANSHIP IN MATHEMATICS
Internal forms of organic cells. Fields of force; their
form; polarity. Effects of surface tension on cell division.
Liquid films: minimal surfaces and figures of equilibrium.
Plateau's experiment. Spiders' webs. Forms of globules,
hanging drops, splashes. Unduloids in the infusoria (e.g.
vorticella); fluted and pleated cells.
Cell aggregates. Surfaces in contact; cell partitions; why
partitions between cells of equal size are plane, and between
cells of unequal size are curved. Tetrahedral symmetry;
hexagonal symmetry. The geometry of the bee's cell and
of beecell architecture. Minimal areas in nature's partition
ing of space.
Spicules and spicular skeletons. Concentric striation in
nature. The fish's age as estimated by the concentric lamel
lation of its scales; compare with the concentric rings in the
trunk of a tree. The skeletons of sponges. The radiolarian
skeleton.
Geodesies. The helicoid geodesic on cylindrical structures
and its purpose; how " stretching tight " and constricting
are effected by fibres arranged in geodesic fashion. The
spiral coil in the trachcal tubes of an insect; the tracheides
of a woody stem.
The logarithmic spiral. Difference between spirals and
helices. The curves of the horns of ruminants, of molluscan
shells, of animals' tails, of the elephant's trunk. The properties
of the logarithmic spiral in its dynamic aspect. Explain
clearly why the molluscan shell, like the creature inside it,
grows in size but does not change in shape: this constant
similarity of form is the characteristic of the logarithmic
spiral. The study of shells generally, morphologically, and
mathematically. The spiral shells of the foraminifera. Torsion
in the horns of sheep and goats. The deer's antlers. The
curvature of beaks and claws.
Phyllotaxis. Spirals. Symmetry.
Shapes of eggs. An egg, just prior to the formation of
its shell, is a fluid body, tending to a spherical shape, en
closed in a membrane. The problem of the shape of the
MATHEMATICS IN BIOLOGY 587
egg: given a practically incompressible fluid, contained in
a deformable capsule which is either entirely inextensible or
only very slightly extensible, and which is placed in a long
elastic tube the walls of which are radially contractile: to
determine the shape of the egg under pressure. At all points
the shape is determined by the law of distribution of radial
pressure within the oviduct; the egg will be compressed in
the middle, and will tend more or less to the form of a cylinder
with spherical ends. From the nature and direction of the
peristaltic wave of the oviduct, the pressure will be greatest
somewhere behind the middle of the egg; in other words,
the tube will be converted for the time being into a more
conical form, and the simple result follows that the anterior
end of the egg becomes broader and the posterior end the
narrower. The mathematical statement of the case is simple.
Comparison of related biological forms. This is a very
large subject, and applies to the whole region of biological
morphology. Basically, it consists of the transformation of
a system of coordinates and a comparative study of the
original and transformed figures (wing, leg, bone, skull, or
what not) in the coordinate system. The new figure in the
transformed system shows the old figure under strain. The
new figure is a function of the new coordinates in precisely
the same way as the old figure is of the original coordinates.
The reader should examine figs. 404, 405, and 406
in Growth and Form (see below), where (i) the outlines of
a human skull are enclosed in a coordinate system of
squares; (ii) the outlines of a chimpanzee's skull are enclosed
in another system determined by points exactly corre
sponding to the intersecting points in the first system. The
new coordinate system though consisting of curved lines
is of a strikingly regular type, and obviously bears a very
simple mathematical relation to the first system. It is for
the biologist to trace the transformed coordinates, for the
mathematician to step in and show the relations between
the new and the old coordinate systems, and then for the
biologist to come in again and explain the relations if he
588 CRAFTSMANSHIP IN MATHEMATICS
can. That the relations are simple, and that they are con
tinuous, are obvious. The logarithmic curve seems to make
its appearance even once more. (See D'Arcy W. Thompson
Chapter XVII.)
The best modern work on the whole subject is Professor
D'Arcy Thompson's Growth and Form. Professor J. Bell
Pettigrew's Design in Nature should also be consulted; it
is a remarkable threevolume work with a multitude of
useful facts and many hundreds of illustrations, but some
of its opinions do not meet with general acceptance.
CHAPTER XLI
Proportion and Symmetry in Art
Proportion, Harmony, and Symmetry
Art is another subject which mathematics gathers into
its ambit, though the mere suggestion is enough to stir the
average artist to anger. But what about perspective? The
subject is generally taught by art teachers as if it consisted
of a number of incomprehensible stereotyped rules. And
what about the geometry underlying design?
But, after all, these things are comparative trifles. Under
lying a great deal of what counts for art is a mathematical
foundation quite unsuspectejjl even by many mathematicians.
The subject is much too farreaching for more than a
few references to it to be made here.
Those qualities in the general disposition of the parts
of a building that are calculated to give pleasure to the
observer are proportion, harmony, and symmetry. In the
dimensions of a building, proportion itself depends essentially
upon the employment of very simple mathematical ratios.
Proportions such as those of an exact cube, or two cubes
PROPORTION AND SYMMETRY IN ART 589
placed side by side, or dimensions increasing by onehalf
(e.g. a room 20 ft. high, 30 ft. wide, 45 ft. long), please the
eye far more than do dimensions taken at random. The
great Gothic architects appear to have been guided in their
designs by proportions based on the equilateral triangle.
By harmony is meant the general balancing of the several
parts of the design. It is proportion applied to the mutual
relations of the details. By symmetry is meant general
uniformity in plan.
Accurate measurements have been made of the Parthenon
and of several of the great cathedrals, and the unvarying
simplicity of the mathematical ratios determining the various
proportions is a very impressive fact. The same thing applies
to natural objects. In particular it applies to the human
figure. An artist does not, of course, measure up a model
before making a selection, but his eye tells him at once if
the proportions are satisfactory. If the human body approaches
anything like perfection, from the crown of the head to the
thigh joint is onehalf the whole height; from the thigh
joint to the kneejoint, from the kneejoint to the heel, and
from the elbowjoint to the end of the longest finger, are
each onefourth of the whole height; from the elbowjoint
to the shoulder is onefifth; from the crown of the head
to the point of the chin is oneeighth. The proportions of a
perfect face are even more remarkable; the ratios of the
distances between the various facial organs, and of the
lengths and widths of the organs, are singularly simple through
out.
In great architecture, even more remarkable than the
linear measurements is the simplicity of the forms of the
various rectilinear and curvilinear spaces. If we analyse
a drawing of, say, the east front of Lincoln Cathedral, we
can discover a series of striking relations amongst the parts.
First enclose it in a rectangle, and then draw the bisecting
vertical line; from the upper end of that vertical, draw
7 pairs of oblique lines, right and left, to the base and sides
of the rectangle, as follows:
590 CRAFTSMANSHIP IN MATHEMATICS
1. Lines (to the angles) which determine both the width
of the design, the tops of the aisle windows, and the bases
of the pediments on the inner buttresses.
2. Lines which determine the outer buttress.
3. Lines which determine the width of the great centre
window.
4. Lines which determine the form of the pediment of
the centre.
5. Lines which determine the form of the pediments of
the smaller gables.
6. Lines which determine the height of the outer but
tress.
7. Lines which determine the height of the inner but
tresses
It will be found that (a) these lines determine the heights
and widths of nearly all the main features of the design,
and (j8) the angles which the obliques make with the hori
zontal are all simple fractions of a right angle. Were the
architect to depart from these simple ratios very appreciably,
the eye would be offended; the trained eye would resent
even a very small departure.
Ratio Simplicity
The key to the harmony of beauty in its more general
sense seems to be the simplicity of the angle relations which
determine or which underline the form of the thing con
sidered beautiful.
It is sometimes said that there are three primary "orders"
of symmetry, viz. those based on the numbers 2, 3, and 5,
respectively. That of the first order is represented by the
half square cut off by a diagonal, that is, the rightangled
isosceles triangle, with angles 45, 45, 90 (angles 1:1: 2);
that of the second order is represented by the half equilateral
triangle cut off by a median, that is, a triangle with angles
30, 60, 90 (angles 1:2:3); that of the third order is
PROPORTION AND SYMMETRY IN ART
591
represented by the half triangle from a pentagram cut off
by a median from the vertex, that is a triangle of 18, 72,
90 (angles 1:4:5). Thus we may easily obtain angle
ratios 1/1, 1/2; 1/2, 1/3, 2/3; 1/4, 1/5, 4/5; and these ratios,
alone or compounded in a simple way, are architecturally
fundamental.
An ellipse, a figure which enters very largely into archi
tectural composition, may be constructed with its principal
diameters of any ratio, but the ellipses which are acceptable
for purposes of symmetrical beauty are those based upon
the simplicity of the ratio of the
angles made by the diagonal with
the sides of the rectangle which
encloses the ellipse. This ratio is
usually one of those mentioned in
the preceding paragraph.
But a much more subtle curve
is the " oval ", better called a
" composite ellipse ", since its axis
is sometimes so short that the oval
ceases to resemble an egg. This
curve has 3 foci (A, B, C in figure),
forming an isosceles triangle. If
from the ends of the base of this
triangle lines be drawn to that end of the axis, D, at the
" flatter " end of the curve, they make an angle which (for
the purposes under consideration) must have a very simple
relation to the angle at the apex of the isosceles triangle
(in the figure, 3 : 1). If this simplicity of ratio is departed
from, the curve is not acceptable as an element of har
monious proportions.
It is a remarkable thing that a form is considered beautiful
when the space which it encloses can be analysed in such a
way that the resulting angles bear proportions to each other
analogous to those which subsist among musical notes.
The basis of musical harmony is that, when two sounds
mingle agreeably, the numbers of vibrations of which they
Fig. 309
592
CRAFTSMANSHIP IN MATHEMATICS
are respectively composed bear a very simple ratio to each
other. All the harmonies are represented by quite simple
fractions, J, , , &c.
Some things to look for in objects considered
beautiful
Boys should be encouraged to take an interest in the
proportions, harmony, and symmetry of beautiful buildings
and other objects. I have known many pupils who claimed
to be beautyblind, really awakened to a new life once they
knew what to look for when examining a thing considered
beautiful a building, a piece of statuary, a picture, a vase,
a piece of Gothic orna
ment.
Search for the beauty
of form of a Greek or
Etruscan vase: it does not
take much finding. Stand
in front, or behind, or at
the side, of the Venus of
Milo in the Louvre; the
particular view matters
little. The extraordinary
beauty of the curves of
the figure, despite their
complexity, imposes itself
Fig. 310 upon the mathematician
whether he will or not. To
the artist the figure is beautiful for reasons which, though ade
quate, he finds a little difficult to explain, or at least difficult
to analyse. The mathematician's approach is an entirely
different approach, but the approach is intensely interesting;
he desires to discover the secret of the artist's construction,
and he sets to work to analyse. Once the artist's secret
stands revealed, his first feeling is one of admiration for
such subtle craftsmanship. The beauty of the thing gradually
PROPORTION AND SYMMETRY IN ART
693
Fig. 3 1 1
grows upon him. The feeling is as much intellectual as
it is emotional, and for that reason it is sometimes of a higher
order than a feeling for beauty that may be
emotional only. The boy who is a failure in
the school studio can nevertheless be taught
what to try to search for in a thing that the
world calls beautiful.
We give figures of (1) an Etruscan vase
showing its component curves and their tan
gential relations; (2) the construction of the
echinus moulding in Greek architecture (note
the 3 composite ellipses); (3) the outline of
the human figure, showing nature's subtle
construction of the sides of the head, neck,
trunk, and outer surfaces of the legs. Note
the tangency throughout. In an analysis of
a perfect human form, these tangential rela
tions seem to persist to the smallest detail.
(Figs. 310, 311, 312.)
If one of the ends of life is the pursuit
of beauty, then mathematics, properly under
stood, is one of the avenues we should
follow. Consider even Einstein's work; what
is its main value? that> underlying the diverse
(K291)
594 CRAFTSMANSHIP IN MATHEMATICS
phenomena of the natural world, there has been dis
covered a harmony more allembracing than any ever before
dreamed of.
I do not know if it may be adequately maintained that
harmony is the most essential factor in beauty, but assuredly
it is the desire for harmony that animates the modern searcher
after the secret of the ordered relations of the universe,
just as it animated the Greeks in their stargazing and their
geometry.
I find it very hard to distinguish the passion for truth
from the quest of beauty. Certainly we need never despair
of the beautyblind boy if he is taught mathematics as it
might be taught.
CHAPTER XLII
Numbers : Their Unexpected Relations
The Theory of Numbers
One noteworthy subject which is lacking in the equip
ment of many of our younger mathematical teachers is the
theory of numbers. Forty or fifty years ago the subject was
included as a matter of course in Sixth Form mathematics.
In those happy days the mathematical work was not narrowed
down to the requirements of the few boys who were going
to read mathematics at the University. The ground then
covered was more extensive, and in many ways was more
interesting. Some schools devoted considerable attention to
the theory of numbers (as the subject is called, though not
very happily), such topics being included as the theory of
perfect, amicable, and polygonal numbers; properties of
prime numbers; possible and impossible forms of square
numbers, of cubes, and of higher powers; the quadratic
NUMBERS: THEIR UNEXPECTED RELATIONS 595
forms of prime numbers; scales of notation; indeterminate
equations; diophantine problems. The magic square and
magic cube were also included. Altogether, the boys were
given an interest in arithmetic and algebra that remained
a permanent possession in after days, and was rarely forgotten.
A final blow was given to this branch of work when the
fiat went forth that circulating decimals being useless and
unpractical, their use must be abandoned. The futility of
circulating decimals in the solution of practical problems
may be granted. But if we ignore them altogether, we cut
off from the learner some of the most striking properties of
numbers; in fact we deny him most of the inner significance
of numerical relations.
I plead for a revival of some of this work, and therefore
indicate a few topics that may be included.
Within the last year or two, a highly competent young
mathematical mistress told me that she had made what
she considered to be a rather striking arithmetical discovery.
Had I come across it before?
This is what she showed me, and then she pointed out a
\ = i4285t
 == 285714
?  428571
57142
J = 714285
f = 57145
few of the wellknown properties of this particular grouping:
the same 6 figures in the same order in all 6 cases; the 1st
group a factor of all the others; the 2nd group a factor of
the 4th and 6th; the 5th group the sum of the 2nd and 3rd;
and so forth. She said quite seriously that although she
had, as a girl, won an open scholarship, she had never during
her school days seen the completed decimal for any one
of these halfdozen simple fractions.
I pointed out that she had not made up the group to the
best advantage, and I modified it thus:
596
CRAFTSMANSHIP IN MATHEMATICS
i
T
3
f
n
r
*
K
T
1
1
4
2
8
5
7
3
4
2
8
5
7
1
2
T
2
8
5
7
1
4
6
T
8
5
7
1
4
2
4
T
6
7
1
4
2
8
5
r
7
1
4
2
8
5
The values of the 6 vulgar fractions may now be read off
either from left to right or from top to bottom. Moreover,
the diagonal lines running upwards from left to right consist
each of the same figure.
Even then she could scarcely credit my statement that
the same principle applied to all circulating decimals what
soever. I suggested she should evaluate the 40 fractions
4~O ITD 4~f t J i > which she did, and made the discovery
that the 40 circulating decimals fell into 8 groups of 5 figures,
all presenting the selfsame symmetry.
1
TT
= 6243$
Tl
0487
3
4T
073lt
4
If
09756
1
TT
= 24396
20
IT
48780
4 1 ~
17073
23
41
5609?
l fl
4T
= 59024
4 T ~
7804&
13
4 1
3170?
2 5
41
60975
18
41
= 43902
:) 3
4 \
0487
2
i 1
70731
.'51
41
75609
3 7
TT
= &)243
4 T ~
7804
:io
4 T
73176
40
41
&7560
fi
IT
= 12195
C,
41 ~
14634
i i
41
2682$
if ~
36585
8
"41
= 19512
14
41
54146
1 li
41
2926
"4T ~
5365
TT
= 21951
17
4T
41465
28
41
68292
2 4
TT
58536
21
4 1
= 51219
10
41
46341
84
41
2926
27
41
65853
39
41
= 95121
20
ll
63414
38
41 ~
&26S2
3 5
TT
5365
Make the boys evaluate these or other similar groups,
and encourage them to search for the curious (though obviously
necessary) relations between the members of each group
and between group and group. For instance, the first decimals
of the above 6 groups are 1:2:3:4:5:6; the sum of the
1st decimals of the 5th and 6th groups is equal to the first
decimal of the 7th group; and so on almost indefinitely. An
JNUMBERS: THEIR UNEXPECTED RELATIONS 597
examination of the numerators of the vulgar fractions gives
the clue to an almost endless number of relations amongst the
40 decimal groups. Clearly all 40 decimals can be evaluated
in 5 minutes; only one actual division is necessary, viz. that
for ^Y. This sort of thing, which applies universally, was
the A B C of Upper Form arithmetic half a century ago.
Suggested Topics
1. Primes and composite numbers; measures and multiples;
tests of divisibility (with algebraic proofs). Familiarity with
the factors of such common numbers as 1001 (=7 X 11 X 13),
of 999 (= 27 X 37), in order to write down at once the factors
of such numbers as 702,702 and 555,888. Eratosthenes 1 sieve.
Fermat's theorem. The number of factors in a composite
number; the number of ways in which a composite number
may be resolved into factors.
2. Perfect numbers. (A perfect number is one which is
equal to the sum of all its divisors, unity included). N =
2 n ~ 1 (2 n 1), the bracketed factor being prime. Examples: 28,
496,8128.
3. Amicable numbers. (Amicable numbers are pairs of
numbers, each member of a pair being equal to the sum of
all the divisors of the other number.) Examples: 220 and
284; 18,416 and 17,296. The formulae are rather long though
easy to manipulate
n 12
I 5 12 22
Fig. 313
4. Polygonal numbers. Teach the pupils (1) to sum
triangular, square, pentagonal, and hexagonal numbers, and
598 CRAFTSMANSHIP IN MATHEMATICS
(2) to find the ?ith term of each series. All are easy and are
full of interest. The necessary figures may be drawn readily.
Pascal's arithmetical triangle.
5. Scales of notation. Illustrate by some of the mediaeval
problems on agetelling cards; weighing with a minimum
number of weights, e.g. binary scale weights 1, 2, 2 2 , 2 3 ,
&c., for one pan; ternary scale weights 1, 3, 3 2 , 3 3 , &c.,
for either or both pans; &c.
6. Congruences. Use Gauss's notation a = b (mod. m)\
e.g. 158 (mod. 7), 36  (mod. 12), 37 = 19 (mod. 6).
Emphasize the fact that a modulus is a divisor. The numbers
15 and 8 are congruent, or they agree, for the modulus 7,
because they agree as regards the divisor 7; they " agree "
in giving the same remainder, 1. The theory of congruences
is necessary for a proper understanding of parts of the theory
of numbers. Very little practice in a few of the more ele
mentary theorems of congruences is required in order to
give necessary facility in subsequent work; e.g.
(1) 72 = 37 = 30 = 9 = 2 = 5 = 12 = 33 (mod. 7).
(2) 100 = 15 (mod. 17).
i.e. 10 2 = 15 (mod. 17),
.'. 10 4 = 15 2 (mod. 17),
.'. 10 4 = 225 (mod. 17),
/. 10 4 = 4 (mod. 17).
From the 225 we have subtracted 13 times the modulus and
have thus brought it down to a number smaller than the
modulus. The pupils should be familiarized with this prin
ciple.
7. Circulating decimals. How many of our younger
readers know how easy it is to write down, almost at top
speed, the complete circulating decimal equivalent to a
vulgar fraction in its lowest terms, if the denominator is
prime, no matter how many figures the period may consist
of? The following section will suffice for a general introduc
tion to this interesting subject.
NUMBERS: THEIR UNEXPECTED RELATIONS 599
Circulating Decimals and Congruences
We will set out below the complete evaluation of the
decimal corresponding to such a fraction, say ^. This gives
a recurring period of 28 places, and we shall therefore write
down 1 followed by 28 ciphers and divide by 29 in the usual
way. We will choose short division in order to show the
successive quotient figures and their corresponding re
mainders clearly. Over the ciphers we will write the successive
numbers 1 to 28, in order to be able to refer at once to any
particular quotient figure or to any particular remainder.
We will call the quotient figures, Q's, and the remainder, R's.
Examine the 28 R's carefully: they consist of all the numbers
1 to 28, the last of them being 1; this typifies all evaluations
of circulating decimals
29 )1000000000000 0000000000000000
Q ' s ~* HSliTTs 27586 2"06 8965517241 37 9 3~1
R s _+ ,. rrn; . 2 , i, * 5 i, . , 20 ,. 28 inrrirT i. *  .rrrr
(It may be observed that the unit figure of each R is identical
with the corresponding Q; but this is not universal; it
occurs only when the main divisor has 9 for its unit figure.)
The pupils' interest may readily be excited in this way:
Twentynine is a rather hard number to divide by. Can
we substitute a smaller and easier number? Yes, at any
rate after we have found the first few figures (Q's) of the
answer by dividing by the 29, say as far as Q n , i.e. when
we have '03448275862. Now begin again at the begin
ning and divide this partanswer by 5 (ignore the decimal
point for the present).
5)3448275862
^0689655, &c.
But these figures 0689655, &c., are the figures beginning at
Q 12 in the answer, and we may continue to divide by 5
600 CRAFTSMANSHIP IN MATHEMATICS
until we reach the end. But note that we no longer bring
down O's, but the Q's we have previously written down.
But we need not have divided by 29 so far as Q n . Suppose
we had gone as far as Q 9 . We write down as before the
figures thus obtained (omitting the decimal point and the
which follows it), prefix a 4, and then divide by 7.
7)^34482768
""62068, &c.
But these figures 62068, &c., are the figures beginning at Q 10
in the answer, and we may continue to divide by 7 until
we reach the end. In this case we have shortened still more
the original division by 29.
But we can shorten the division by 29 still further. Sup
pose we had proceeded as far as Q 3 , and had obtained 034.
As before we write down the figures thus obtained (omitting
the decimal point and the cipher), prefix a 4, and then
divide by 9.
9) 4 34
_48, c.
But these figures 48, &c., are the figures beginning at Q 4
in the answer, and we may continue to divide by 9 until
we reach the end. Thus we may begin to divide by 9 after
obtaining only 3 figures by dividing by 29.
Note carefully how we proceed with the division in the
last case. We had, to begin with, 034. To proceed with the
division we prefix a 4 to the 3, put each new Q in its proper
place, and remember to " bring it down " (not bring down
a 0) when its turn comes.
9's into 43 = 4 and 7 over. Hence 0344.
9's 74 = 8 2 03448.
9's 24 = 2 6 034482.
9 s 68 = 7 5 0344827.
Really, however, it is unnecessary to divide by 29 more
than 2 places, i.e. when we have obtained Q x and Q 2 (= 03).
This time we prefix to the Sal, and thus obtain 13 for our
NUMBERS: THEIR UNEXPECTED RELATIONS 601
initial bit of new dividend. This time we can use the easy
divisor 3.
3's into 13 = 4 and 1 over. Hence 034.
3>s 14 = 4 2 0344.
3>s 24 = 8 03448.
3' s 08 = 2 2 034482.
3's 22 = 7 1 0344827.
Observe carefully in this case that, at each step, the new
quotient figure and the over figure give, reversed, the number
to be divided at the next step; e.g. 4 and 1 over in the first
line give 14, the number to be divided in the second line.
And so generally.
To divide by 3 is so easy that we may evaluate the whole
28 figures of the period in half a minute.
But we need not divide at all. We may begin at the other
end, and multiply instead.
Suppose we know the last 5 figures, . . . 37931 (Q 24
to Q 28 ). We may multiply the end figure 1 by 11 and obtain
Q 23 , and then proceed in this way:
IXll = 11; 1 down (= Q 23 ) and 1 to carry.
Hence 137931.
(3 x 11) + 1 = 34; 4 down (= Q 22 ) and 3 to carry,
Hence 4137931.
(9 X 11) + 3 = 102; 2 down (= Q 21 ) and 10 to carry,
Hence ..24137931.
But other multipliers might be used. Suppose, for instance,
we know only the very last figure, 1 (= Q 28 ). This is quite
enough, if we use the multiplier 3, and we may finish the
whole thing in a few seconds.
1x3 =3 31.
3x3 =9 931.
9x3 =27 7931.
(7 x 3) + 2 = 23 37931.
(3 x 3) + 2= 11 ...137931.
A similar scheme applies universally. Whence the secret?
602 CRAFTSMANSHIP IN MATHEMATICS
The secret lies in the R's and in the Q's, and in the use of
congruences.
Give the pupils a ten minutes' lesson on congruences.
The Sixth Form specialists are almost certain to hit upon
the solution. The following hints ought in any case to
suffice.
Consider another example:
17) 10000000000000000
Q's > 0588235 2 oTTT? 647
R's > " " 1 i. . . . >. n . i
The divisor 17 may be regarded as the modulus of a
congruence.
(i) 10 1 = 10,
.. (10 1 ) 2 = 10 a , i.e. 10 2 = 100  15 = R 2 .
(ii) 10 l = 10, and 10 2 = 15;
/. 10 1 .10 2 = 10.15, i.e. 10 3 = 150 = 14 = R 3 .
(iii) 10 2 = 15, and 10 3 = 14;
.'. 10 2 .10 3 = 15.14, i.e. 10 5 = 210 = 6 = R 5 .
(iv) 10 3 = 14, and 10 5 = 6;
/. 10 3 .10 5 EE 14.6, i.e. 10 8 = 84 = 16 = R 8 .
All these results agree with the actual division. Clearly,
then, when the first R has been found by actual division,
any other remainder whatsoever may be found by applying
the principles of congruences to the powers of 10.
It thus follows that once we have detected a multiple
relation between a pair of R's, the multiple may be used as
a general divisor for obtaining Q's, and actual dividing in
the original division need be carried only a very little distance
Consider another example:
59)1000000000000, &c.
Q's> 16949152542,~&T.
n >g ^ 10 41 it 29 64 <1 1C IS It iTfl C
An examination of the R's shows that R 5 = 6R 6 . This
gives us the clue to the relation between every pair of succes
NUMBERS: THEIR UNEXPECTED RELATIONS 603
sive R's, and therefore to every pair of successive Q's. For
instance,
R 5 = 6R 6 (54  6 . 9)
R 6 f 3M = 6R 7 (9 + 177 = 6 . 31)
R 7 + M = 6R 8 (31 + 59 = 6 . 15)
R 8 + 3M  6R 9 (15 f 177  6 . 32),
where the multiple relation suggests the divisor 6.
Consider the first of these, R 5 = 6R 6 . If we have pro
ceeded with the division as far as Q 6 , we divide Q 6 by 6,
and so obtain Q 7 (= 1) and 3 over; 6's into 31 = 5 and
1 over; &c.
Or, consider the last of the 4 relations: R 8 + 3M = 6R 9 .
If we have proceeded with the original division as far as
Q 9 , we prefix a 3 (representing 3M) to Q 9 (= 2), making
32, divide by 6, and so obtain Q 10 .
Or, consider the third of the 4 relations: R 7 + M = 6R 8 .
If we have proceeded with the original division as far as Q 8 ,
we prefix a 1 (representing 1M) to Q 8 (=5), making 15,
divide by 6, and so obtain Q 9 .
We may begin to divide by 6 at any point after Q . It
is merely a matter of prefixing a figure indicated by x in
In our first example,  2 ^, we first divided by 5, because
we noted the relation R x == 5R n ; then we divided by 7,
because we noted the relation R 3 7R n ; then we divided
by 9, because we noted the relation R 9 = 9Rn An easy
divisor can always be obtained by examining the R's in this
way. We finally divided by 3 because we noted that R 10 = 3R U .
But with a little practice we can dispense with the R's
altogether, and detect a multiple relation amongst the early
Q's, these being obtained, of course, by actual division.
For instance
^ = 6434782608695652173913.
After obtaining about 5 Q's, we might notice that by pre
fixing 2 to Q 2 , making 24, we might divide by 7 and obtain
Q 3 , and so on continuously.
604 CRAFTSMANSHIP IN MATHEMATICS
Here is an example for the boys to complete. The recurring
period consists of 646 places. A good Sixth Form boy
ought to write down 1 figure per second, and so do the whole
thing in 10 or 11 minutes.
F J T = 601545595 . . . 057187011
A suitable divisor or multiplier is 11. The division by 11 may
be begun after the first 3 places are obtained (001) by
prefixing 6 to Q l and so obtaining 6001. (A smaller divisor
may soon be found.)
11 's into 60 = 5 (= Q 4 ) and 5 over;
ll's 50 = 4(^Q 5 ) 6 ;
ll's 61^5(=Q 6 ) 6 ;
ll's 655(=Q 7 ) ,,10 ; &c.
In the multiplication,
7 (= Q 6 4 6 ) X 11  77; 7 (= Q 643 ) and carry 7;
{ 1 (= Q 6 4 5 ) X 11 } + 7 = 18; 8 (= Q 642 ) and carry 1, &c.
The result is easily checked by selecting other divisors.
If the numerator of the vulgar fraction is other than
unity, the equivalent decimals will consist of the same figures
as when the numerator is unity, and they will be in the same
order, but the period will begin in a different place, easily
discovered. But the subject, which is a source of delight
to most boys, cannot be carried further here.
Magic Squares and Magic Cubes
Can time spent on this subject be justified? If as a
mathematical topic for purposes of formal teaching, no. If
as a subject for creating a lasting mathematical interest in
the less mathematically inclined boys, yes.
Magic squares have interested the greatest mathematicians.
The wonderful harmony and symmetry of the numbers so
grouped have always tended to attract their attention. Boys
are always impressed by the mysterious regularity that
NUMBERS: THEIR UNEXPECTED RELATIONS 605
emerges in so many ways when they study magic squares.
Three or four lessons on the subject are well worth giving,
though in so short a time the secrets of the construction of
some of the remarkable squares that have been constructed
by mathematicians could not be given.
The simplest and bestknown construction (for a square
with an odd number of sides) is the following: the method
5
4
10
3
9
15
2
8
14
20
1
7
13
19
25
G
12
18
24
II
17
23
16
22
21
3
16
9
22
15
20
8
21
14
2
7
25
13
1
19
24
12
5
18
6
II
4
17
10
23
e>
c
Fig. 3H
underlying the symmetrical transfer of the numbers in the
temporary outer cells is obvious. Horizontally, vertically,
and diagonally, the sum of the numbers is 65.
Paste such a magic square round a roller, the circumference
of which is equal to AB or BC. Two squares should be pre
pared, one to be rolled round from side to side, so that AB
coincides with DC, and one from top to bottom so that
AD coincides with BC. The consecutive numbers in the
various diagonals of thfe rolled up squares give the learner
the real secret of the construction of the simpler squares.
The famous Benjamin Franklin was the inventor of magic
squares with properties that always fill the learner with
astonishment. The construction of his 8 X 8 and 16 X 16
squares is quite simple, and this very simplicity goes far
to create the astonishment. They are to be found in all the
textbooks on the subject, and every boy should know them.
606
CRAFTSMANSHIP IN MATHEMATICS
The problem of finding the number of different ways
in which the numbers, say, 1 to 25, may be arranged in a
square is worth looking into, though no general solution
has yet been discovered. Boys find out at once, of course,
that variations are easy to make and are numerous.
Not all the textbooks point out the device for making
magic squares so that the products of the columns and rows
Fig. 315
are constant. It is simply a question of using the numbers
in the ordinary squares as indices of some selected number
for the new square. We append the usual 3x3 square
(common sum 15) and one of its cousins (common product
= 2 15 = 32768).
" Magic cubes " may be touched upon, say 3 X 3 X 3
(1 to 27). The sum of the numbers in each row is 42, not
only in each face shown, but through the faces, front to back;
also the diagonals of the cube as well as the diagonals of
some of the squares.
Magic circles, pentagrams, &c., are hardly worth spending
time over.
The best books on the subject are (1) Magic Squares and
NUMBERS: THEIR UNEXPECTED RELATIONS 607
Magic Cubes, John Willis: (2) Magic Squares and Cubes,
W. S. Andrews; (3) Les Espaces Arithmdtiques Hypermagiques,
Gabriel Arnoux; (4) Les Carres Magiques, M. Frolow;
(5) Le Probleme d'Euler et les Carres Magiques, Atlas, M.
Frolow.
Magnitudes. Great and Small
When attempting to help a boy to form a clearer conception
of the significance of very large numbers, say those con
cerning stellar distances or atomic magnitudes, it is essential
for the teacher to eliminate from the problem every kind
of avoidable complexity. To form a conception of a great
number is quite difficult enough in itself, and to a boy the
difficulty may prove insuperable. On one occasion I heard
a teacher attacking our old friend the " lightyear ", in
favour of its new rival the " parsec ", simply on the ground
that the latter made astronomers' computations easier. Now
the lightyear is a perfectly well understood thing. In
mechanics we often define distance as the product of velocity
and time (s = vt), as every child knows; and we apply
this selfsame principle to the distance known as a light
year, the new unit being determined by the product of the
velocity of light (miles per second) and the number of
seconds in a year. But the parsec is the distance corre
sponding to the parallax of 1", and a simple calculation
shows that it is 3*26 times as long as the lightyear; and
this trigonometrical method of determining star distances
compels the learner to think in terms of the semimajor
axis of the earth's orbit. The complexity is entirely un
necessary in school work; it tends to obscure the main
thing the boy is supposed to be thinking about.
If the learner already knows that the velocity of light
is 186,000 miles a second, simple arithmetic tells him that
the length of the lightyear is, approximately,
(186,000 X 60 x 60 X 24 x 365) miles, i.e. 6 x 10 12 miles,
or 6 billion miles. Thus, when the learner is told that a Cen
608 CRAFTSMANSHIP IN MATHEMATICS
tauri is 4 lightyears distant, he knows that this means 24
billion miles; and that the 1,000,000 lightyears representing
the probable distance of the remoter nebulae is a distance
of 6 X 10 18 (six trillion) miles. Or, he may be told that
the mass of the H atom is 166 X 10~ 24 grams, when he
sees at once that 10 24 H atoms together must weigh 1 grams.
But are these vast numbers anything more than mere
words to the boy? What does a quadrillion signify to him?
or even a trillion or a billion? or even a million? Is it of
any use to try to make the boy realize the significance of
such numbers? or just to leave them as mere words? or
not to mention them at all and merely to give some such
illustration as Kelvin's earthsized sphere full of cricket
balls?
I have tried the experiment of giving to boys such illus
trations as these: (1) the number of molecules in 1 c. c. of
gas is about 200 trillions (2 X 10 20 ), a number equal to the
number of grains of fine sand, 70,000 to the cubic inch,
in a layer 1 foot deep, covering the whole surface of England
and Wales; (2) the number of molecules in a single drop of
water is about 1700 trillions (17 x 10 21 ), a number just about
equal to the number of drops of water in a layer 7 inches
deep completely covering a sphere the size of the earth.
But I have always found that such illustrations merely
give rise to vague wonderment. The pupil himself makes
no personal effort to realize the magnitude of the numbers;
and this is fatal.
Such an effort is indispensable. The best plan, perhaps,
is to make the pupil first consider carefully the magnitude of
a million, then of a billion, a trillion, a quadrillion, successively
(10 6 , 10 12 , 10 18 , 10 24 ). For instance, an ordinary watch ticks
5 times a second or 1000 times in about 3 minutes, or a
million times in about 2 days and 2 nights. Let this fact be
assimilated as a basic fact, first. Now let the boy think about a
billion. Evidently, a watch would take (for the present
purpose, all the underlying assumptions may be accepted)
about 6000 years to tick a billion times (2 days X 10 6 ), so
NUMBERS: THEIR UNEXPECTED RELATIONS 609
that if a watch had started to tick at the time King Solomon
was building the Jewish temple, it would not yet have ticked
half a billion times. Now proceed to a trillion, and then
to a quadrillion. Evidently the watch would take 6000
million years to tick a trillion times, and 6000 billion years
to tick a quadrillion times. An approach of this kind to
the subject does not take long, and a boy fond of arithmetic
may be encouraged to invent illustrations of his own. It
is worth while. It is worth one's own while, if the attempt
has never been made before. It is, indeed, hard to realize
the significance of the statement that lightwaves tap the
retina of the eye billions of times a second. Yet how are
we to escape accepting this frequency if we accept the measured
velocity of light and the measured length of lightwaves?
Impress upon the boys the fact that the inference is inescap
able.
The description of the manufacture of such a thing as a
diffraction grating with lines ruled 20,000 to the inch, or of
Dr. J. W. Beams* mechanical production of light flashes of
only 10~ 7 second duration, serves to impress pupils with the
sense of reality of small things.
Boys must understand that both stellar magnitudes and
atomic magnitudes are, for the most part, calculated values
and not directly measured values, and that the calculations
are, in the main, based on inferential evidence, the inferences
being drawn partly from known facts, partly from hypotheses.
But converging evidence of different kinds justifies a feeling
of confidence in the probability of the truth of the estimate.
So much so is this the case, that the natural repugnance of
the mind to accept statements which seem to be so contra
dictory of everyday experience, and therefore to " common
sense ", is overcome. Still, the nature of the evidence avail
able must be borne in mind. So must the amazing nature of
the results.
(E 291 ) 40
610 CRAFTSMANSHIP IN MATHEMATICS
The same subject. Sir James Jeans 9 Methods
Sir James Jeans, in his two recent books, The Universe
Around Us and The Mysterious Universe, has adopted
various devices for helping the mathematically uninitiated
to realize the significance of large numbers. The first book
is " written in simple language " and is intended to be
" intelligible to readers with no special scientific attain
ments ". The second book " may be read as a sequel ".
In order to find out how the books might appeal to some
of their readers, I induced seven nonmathematical friends
(two specialists in Classics, two in History, one in Modern
Languages, two in Science) to read the books through and
then submit to be questioned on the meaning of the following
(and a few other) extracts:
1. " Less than a thousand thousand millionth part."
2. " 15 million million years."
3. " 2000 million lightyears. "
4. " The nearest star, Proxima Centauri, is 25,000,000
million miles away."
5. " An average star contains about 10 56 molecules."
The chemist was on the spot at once. The biologist had
to do a good deal of thinking, but he got there at last. But
the other five? They failed utterly to understand what the
numbers signified, though all when at school had taken
mathematics in the School Certificate (or its equivalent),
and one in the Higher Certificate. " What is the difference
between ' 15 million million years ' and ' 15 million years '?"
" Oh, the former means just a few more millions than the
latter, I suppose." To the Higher Certificate man I said,
" Compare the increase of 10 4 to 10 7 with the increase 10 53
to 10 56 ." He replied: " It is exactly the same thing, for in
each case you have increased the index by 3. If you take
10 4 from 10 7 you get 9,990,000, so that if you take 10 53 from
10 56 the difference must be just the same "! And so generally.
The five examinees had but the vaguest notions of what the
NUMBERS: THEIR UNEXPECTED RELATIONS 611
numbers meant. Finally they all admitted that, elementary
as the mathematics of the books appeared to be, they simply
did not understand it. The books had left in their mind
feelings of intense wonderment, but the real facts they had
not grasped at all. It would be interesting if other teachers
would test some of their own friends similarly.
I am not sure that we gain anything by writing "15
million million " instead of " 15,000,000,000,000 ", or " 15
billion ", or " 15. 10 12 ". Are the words " million million "
significant?
Some of Sir James Jeans' illustrations are well worth
examining. Here is one:
1. Age of telescopic astronomy, 300 years.
2. Age of astronomical science, 3000
3. Age of man on earth, 300,000 *
4. Age of life on earth, 200,000,000
A teacher can make much of such a comparative device,
especially if it is illustrated by a timeline.
Here is a second:
The earth's orbit is 600,000,000 miles. Represent this
by a pinhead jV in diameter. Then,
1. Sun is 1/3400" in diameter.
2. Earth is 1/340,000" in diameter (invisible under the
most powerful microscope).
3. Nearest star is 225 miles away.
4. Nearest nebula is 30,000 miles away.
5. Remotest nebulce are 4,000,000 miles away.
A third:
The average temperature of the sun's interior is 50,000,000
degrees (it is probably very much higher). Think of a pinhead
of matter of this temperature. It would require the energy
of an engine of 3000 billion horsepower to maintain it. The
* Possibly much longer than this.
612 CRAFTSMANSHIP IN MATHEMATICS
pinhead would emit enough heat to kill anybody 1000 miles
away.
This illustration is less successful than the others. There
is no gradation. A radius of 1000 miles represents an area
nearly as large as Europe, but to say that a pinhead of matter
would be so hot as to kill off the whole population of Europe
does not leave on the mind a sufficiently definite mathematical
impression. A succession of preparatory stages is desirable.
To me, however, the temperature in question is utterly un
imaginable; I cannot get much beyond the mere arithmetic,
though I am fairly familiar with the highest temperatures
that have been produced artificially.
A fourth:
For IQQ of its journey, the light by which we see the
remotest nebulae travelled towards an earth not yet inhabited
by man; yet the radius of the universe is 14 times as great
as the distance of the remotest nebulae. This is obviously a
better way of bringing home the fact than by giving the length
of the radius of the universe in miles (2000 million lightyears
= 12,000 trillion miles), though if this number were given
it would add to the boy's interest to tell him that this
number is roughly comparable to the number of molecules
in a single drop of water.
In dealing with large numbers, it is a sound teaching
principle (1) to illustrate them by diagrams of some sort,
(2) to approach them by stages; and it is a simple matter
to show the boys how these things may best be done. There
is probably no better plan than that of a succession of distance
or time lines drawn to gradually diminishing scales. The
boys will have gleaned the main idea from their history
lessons. Here is a skeleton history timeline, 55 B.C. to
A.D. 1931:
S5&G. 410 1066 J588 1815 1931
NUMBERS: THEIR UNEXPECTED RELATIONS 613
Let the line A represent 10 units; B, 100; C, 1000.
o
B _
c .
i
*_
10
2 3
4
5
6
7
8
9
10
20 JO
40
50
60
7O
80
90
100
100 200 500 K)0 500 600 700 800 900 1000
Point out how A has shrunk to yjy of itself in B and to y^o
of itself in C. The first line might represent a tenyearold
boy's 10 birthdays; the second line would then show the
boy's life history in ^ the length of the first, and C in T ^Q.
Develop this general idea carefully.
Now deliberately set a trap that will catch 90 per cent of
the class:
" Let us devise a number line the successive parts of
which shall represent the comparative sizes of really big
numbers. You know that 10 6 , 10 12 , 10 18 , 10 24 represent,
respectively, millions, billions, trillions, quadrillions.
A B
C
o
e
fc io 6
IO U
id 18
16"
^ Millions !
1
\
1

'* r BtUiorts
1
1
1^ Trilli
^QJ^5 _ j
1
1
1
1
"Is this allright?" "Yes." " Caught. Surely if
AC represents a billion it must be a million times as long
as AB, and AD a million times as long as that. If AB is
1 inch, AC must be made 16 miles long; AD, 16 million
miles; and AE 16 billion miles." And so on.
Books to consult:
1. Thdwie des Nombres, Desmarest (Hachette).
2. Theory of Numbers, Peter Barlow (an old book, but still very
suggestive).
6U CRAFTSMANSHIP IN MATHEMATICS
CHAPTER XLIII
Time and the Calendar
This subject is essentially mathematical, and it should
be the business of the mathematical staff to see that the
following topics are included in their scheme of instruc
tion.
1. Greenwich mean time. How the length of the solar
day is affected, (a) by the variable movement of the earth
in its orbit; (j8) by the fact that the axis of the earth is not
perpendicular to the plane of the orbit.
2. The modern clock. The astronomical timekeeper is
a " free " pendulum swinging in a vacuum chamber. How
its swing is maintained, and how the pendulum of the " slave "
clock (which does the work of moving the hands round tbf^
dial) is made to swing synchronously with it.
3. Sidereal time; its significance and use.
4. Summer time; opposition to its adoption. The legal
definition.
5. Zone standard times for different countries. How the
zone of other countries differs from Greenwich time by an
integral number of hours. Why 5 standard times in U.S.A.
and Canada and why 3 in Brazil.
6. The Date or Calendar line. Let the boys examine a
good Mercator map and discover for themselves how the
line differs from the 180th meridian. Boys are often
puzzled about the reason for different days, say Monday
and Tuesday, on the two sides of this line. Let them
think the thing out for themselves. The practical difficulty
of running the line through a group of islands instead of
round them.
7. Calendar problems. Successive reforms of the calendar.
Opposition to further reform religious and social, not scientific.
Why not 13 months of 28 days each, and one noncalendar
TIME AND THE CALENDAR 615
(2 in leap year) day during the year? or some other scheme
of a more even division than at present? Why even a
fixed Easter is opposed. How Easter is determined for
each year. The League of Nations and the Reform of the
Calendar.
8. The history of timemeasuring. Clepsydras, sand clocks,
graduated candles, sundials, clocks.
CHAPTER XLIV
Mathematical Recreations
The multitude of problems usually classified under this
heading may be made a very serious factor of the mathematical
course. The average boy will face a good deal of drudgery
if it is a question of solving a puzzle, or seeing his way
through a trick, or liberating himself from a trap. The
majority of the socalled mathematical recreations may be
grouped around definite mathematical principles; if they
are thus grouped, if the underlying principle of a group is
thoroughly mastered, and if the members of the group are
treated as applications of the principle, the work becomes
as serious as it is interesting. There is no better means of
giving a boy a permanent interest in mathematics than to
help him to achieve a mastery of the commoner forms of
mathematical puzzles and fallacies. A few principal topics
may be suggested:
1. Arithmetical puzzles, especially those from mediaeval
sources.
2. Geometrical problems and paradoxes. Shunting and
ferryboat problems. Paradromic rings.
3. Chessboard problems.
4. Unicursal problems. Mazes.
616 CRAFTSMANSHIP IN MATHEMATICS
5. Playingcard tricks. } Only partly mathematical,
6. Cats' cradles. \ but the necessary analy
7. Ciphers and cryptographs, j sis is instructive.
8. Algebraical and geometrical fallacies.
An isolated problem like Kirkman's schoolgirls problem is
also well worth doing, if only for the patient analysis that a
solution of the problem demands.
Nearly all the necessary material may be found in the
late Mr. Rouse Ball's Mathematical Recreations, but the
literature of the subject is extensive. Every mathematical
teacher should have on his shelves the works of fidouard
Lucas; they include everything of interest. The late Henry
Dudeney's books are also useful.
I append a manageable figure (it is new) to illustrate the
principle of Captain Turton's geometrical fallacy, one of
Fig. 317
the best I know. Be it remembered that in all cases like this
the figure is the one thing that matters, if the fallacy is to be
well concealed.
MATHEMATICAL RECREATIONS 617
ABC is an isosceles A of 45, 45, 90.
Draw CD equal to CA and J. BC.
Bisect AB in E, join DE and produce to F in CB produced.
Bisect AF in G and DF in K. Draw GH FA, and KH
1 FD, meeting in H. Join FH, AH, DH, CH.
AFGHAAGH, /. FHAH;
AFKH = ADKH, /. FH = DH;
Hence AH = DH.
In the As ACH, DCH,
CA = CD (constr.),
AH  DH (proved),
CH is common,
/. the As are congruent;
 ZDCH.
Take away the common angle FCH;
/. ZACF ZDCF,
i.e. 45 = 90, or 1 = 2.
CHAPTER XLV
Non Euclidean Geometry
What does " nonEuclidean " Mean?
My experience of the teaching of nonEuclidean geometry
has been slight (not more than 3 or 4 lessons in all) and
not very encouraging, though in all cases the teachers were
certainly competent and the boys (Sixth Form specialists)
able and wellgrounded. Nevertheless, the opinion of many
prominent mathematical teachers is that boys ought to
618 CRAFTSMANSHIP IN MATHEMATICS
know something about the subject. Personally I think it is
too difficult, and is best taken at the University later.
It is essential for a teacher who decides to include the
geometry in his Sixth Form course, first to familiarize himself
both with the whole subject and with its implications. Here
is a possible first course of reading:
1. The controversies of the last hundred years concerning
Euclid's parallel postulate.
2. Some such book as Hilbert's Foundations of Geometry.
3. Part II of Poincare's Science and Hypothesis.
4. Mr. Fletcher's article in No. 163 of the Mathematical
Gazette, viz. " A method of studying nonEuclidean
geometry ".
Captain Elliott writes a suggestive short article on " Practical
nonEuclidean Geometry " in No. 177 of the Gazette.
At the beginning of the nineteenth century, almost
simultaneously, Lobatscheffski, a Russian, and Bolyai, a
Hungarian, showed irrefutably that a proof of the parallel
postulate is impossible. It will be remembered that Euclid
himself seemed to recognize a difference in the degree of
conviction carried to the mind by his statement concerning
parallels, compared with that of his other fundamental assump
tions; and he called the statement a postulate rather than
an axiom. It is incorrect to include the statement as his 12th
axiom, as is commonly done.
Lobatscheffski assumed that through a point an infinite
number of parallels may be drawn to a given straight line,
but he retained all the other basic assumptions of Euclid.
On these foundations he built up a series of theorems which
are perfectly selfconsistent and noncontradictory; the
geometry is as impeccable in its logic as Euclid's. The
theorems are, however, at first sight disconcerting; for
instance, (1) the sum of the angles of a triangle is always less
than two right angles, and the difference between that sum
and two right angles is proportional to the area of the triangle;
(2) it is impossible to construct a figure similar to a given
NONEUCLIDEAN GEOMETRY 619
figure but of different dimensions. Lobatscheff ski's pro
positions have little or no relation to those of Euclid, but
they are none the less logically interconnected. Let the
reader try to reconstruct, say, the first 32 propositions of
Euclid, Book I, on the assumption that the parallel postulate
(the " 12th axiom ") is untenable; he will probably be more
than a little surprised.
Riemann, a German mathematician, likewise rejected
Euclid's parallel postulate. He also rejected the axiom that
only one line can pass through two points. Otherwise he
accepted Euclid's assumptions. The system of geometry
which he then built up does not differ essentially from spherical
geometry. On a sphere, through two given points, we can
in general draw only one great circle, the arc of which between
the two points therefore represents the shortest distance,
and hence the straightest line between them. But there is
one exception; if the two given points are at the ends of
a diameter, an infinite number of great circles can be drawn
through them. In the same way, in Riemann's geometry,
through two points only one straight line can in general
be drawn, but there are exceptional cases in which through
two points an infinite number of straight lines can be drawn.
Riemann's " space " is finite though unbounded, just as
the surface of a sphere is finite though unbounded.
Thus Lobatscheffski's and Riemann's geometries, though
both nonEuclidean, were, in a measure, opposed to each
other.
In Euclid's geometry, the anglesum of a triangle is two
right angles, in Lobatscheffski's less than two right angles,
in Riemann's greater than two right angles.
In Euclid's geometry, the number of parallel lines that
can be drawn through a given point to a given line is one;
in Lobatscheffski's, an infinite number; in Riemann's, none.
Euclidean geometry (which includes all school geometries)
retains the parallel postulate; nonEuclidean geometries are
those which reject the postulate.
620 CRAFTSMANSHIP IN MATHEMATICS
Which Geometry is True?
, Mr. Fletcher asks the question, which geometry is true?
and answers it by saying that they are all true, though the
whole question turns on the nature of a straight line. The
straight line, being elementary and fundamental, cannot,
however, be " defined ", for there is nothing simpler in
terms of which it can be expressed. Thus we are driven
to an indirect definition by axioms. But inasmuch as the
axioms of the three geometries differ, it is obvious that
they define different things. But by " things " we mean
not material objects but ideas suggested by them. All three
geometries are true but only as applying to the " things "
known or unknown to which they refer.
But when it comes to the application of geometry to the
facts of the external world, the question is, as Mr. Fletcher
points out, different. " Now we have to ask: which of these
absolute pure sciences applies the most conveniently or the
most exactly to the facts with which we are dealing? In large
scale work, where alone the differences in the results of
the geometries are large enough to be apparent, we are dealing
chiefly with the form of a ray of light, or the line of action
of gravitation. It is easy to see that these forms may differ
according to circumstances; that while, as seems now to be
probable, Euclid's geometry may be applicable with all
necessary exactness to those rays * at an infinite distance '
from gravitating matter, Lobatscheff ski's, or more probably
Riemann's, may afford a better tool for dealing with them
in the neighbourhood of such matter."
Since in the material world in which we live, Euclid's
parallel postulate seems to be satisfied, Euclid's geometry
will continue to be the geometry of practical life and hence
of our schools.
" The essential requisite for clear thinking on the* subject,"
Mr. Fletcher says, " is the maintenance of the distinction
between the pure science and the applied. The * things '
with which the former deals are ideas, abstractions; it can
NONEUCLIDEAN GEOMETRY 621
only proceed from axioms, but on that basis its results
are absolute. The latter deals with * facts ', with * external
things '; its basis is experimental, and its results approximate. "
Impress upon Sixth Form specialists that in our ordinary
geometry we always argue as if we were living on a plane,
whereas really we are living on a sphere. The surface of the
very table we write on is, strictly, part of the surface of a
sphere of about 4000 miles radius. Practically, it is a plane,
of course, but if we build up a theoretical system (as Euclid
did) on the assumption that we are dealing with actual planes
instead of with parts of a spherical surface, how can our
system be free from possible fallacy? And if we apply that
system to the measurement of stellar distances, how can we
logically assume that it is strictly applicable?
Relativity
It is not a difficult matter to give boys a clear understanding
of the special theory * of Relativity, but the general theory
is much too difficult for them. Even so, it is possible to
give them one or two comparatively elementary lessons on the
general theory, to enable them to see that the final acceleration
difference in Newton's and Einstein's gravitation formulae,
unlike as these formulae are in appearance, is almost insigni
ficant. Do not confuse " spacetime " with hyperspace, a
totally different thing. Spacetime is merely a mathematical
abstraction devised to meet the indispensable need of consider
ing time and threedimensional space together. The greater
part of Professor Nunn's admirably written book on Relativity
can be understood by Sixth Form specialists, as I know from
experience.
Hyperspace
When we come to the question of hyperspace, we are in
a region of difficulty too great for all but the very exceptional
* The Relativity of Simultaneity is apt to be a little puzzling. For suggestions
see Science Teaching, pp. 35772.
622 CRAFTSMANSHIP IN MATHEMATICS
boy. Even some professional mathematicians still have deeply
rooted prejudices against Ndimensional space. The crudest
form of prejudice is what may be called the " common
sense " opinion that as space cannot have more than three
dimensions, any consideration of hyperspace is obviously
nonsense. When Einstein announced his general theory,
a distinguished Oxford philosopher wrote an indignant
letter to The Times, pointing out (amongst other things) that
inasmuch as Aristotle himself had pronounced space to be
just long and broad and deep, in other words threedimen
sional, there was nothing more to be said: Einstein's irrever
ence was almost unpardonable. Evidently the writer of that
letter was under a complete misapprehension as to the nature
of Relativity. Einstein's space is not fourdimensional but
threedimensional, though cosmically it is probably not quite
homaloidal but slightly curved. Einstein's fourth dimension
is time, not space. The same writer also probably misunder
stood the nature of geometry as a science. Geometry certainly
did start as a form of " earthmeasuring ", but even in the
time of the ancient Greeks it had developed into a semi
abstract science, to be deduced from a limited number of
axioms and definitions. For more than 2000 years after
Euclid, it was supposed that axioms were selfevident truths
about the real world. Only one axiom, that concerning
parallels, fell short of the high standard of the others: it
was not selfevident. Attempts to prove it all failed, and
at last it was realized that a logical system of geometry could
be constructed by starting with the denial of the axiom, or
postulate as it ought to be called.
Students of Relativity need not concern themselves much
with hyperspace, but it must not be thought that hyper
geometry can have no application to the geometry of the real
world or to physics. Beginners in wave mechanics naturally
assume that the three dimensions required in Schrodinger's
theory of the motion of a single particle are the three
dimensions of ordinary space, but as soon as we come to two
particles six dimensions are required. Many problems in
NONEUCLIDEAN GEOMETRY 623
thermodynamics require a number of dimensions exceeding
three, though perhaps " degrees of freedom " rather than
" dimensions " is a term more acceptable to some people.
Books to consult:
1. Einstein's Nottingham University Lecture of June 6, 1930.
(See Nature for June 14.)
2. Professor Forsyth's address, Dimensions in Geometry y to the
Mathematical Association. (See Gazette, 212.) This address is
most illuminating.
3. Professor Sommerville's Geometry of N Dimensions.
4. Professor Baker's Principles of Geometry, Vol. IV.
(The last two books should be read by teachers of mathematics,
but, of course, for advancing their own knowledge, not for actual
teaching purposes.)
CHAPTER XLVI
The Philosophy of Mathematics
Mathematical teachers will be well advised to admit
that the philosophical foundations of mathematics is a subject
which is outside the limits of Sixth Form work, save in the
case of very exceptional boys. There are, however, a few
points of a sufficiently simple character that can be included,
if only in order that boys may, before leaving school, lose
some of the " cocksureness " that early mathematical success
so often excites, and learn that mathematical truth is, after
all, something that is still far from being absolute, something
that is still relative.
Mathematical and other Reasoning
One or two formal lessons on the general nature of
reasoning, deductive and inductive, are advisable. This does
not mean that time should be spent on formal logic, except
in so far as is necessary for a clear understanding of the
624 CRAFTSMANSHIP IN MATHEMATICS
syllogism, and that is necessary. The mere settingout of
mathematical truth, as distinguished from the search and
discovery of it, is essentially syllogistic, synthetic, and deduc
tive in character. But mathematical reasoning is not deductive;
it is above all things analytical, inductive.
Let the boys understand clearly that the elementary
scheme of syllogistic reasoning that at one time passed for
" logic ", the formal deductive logic of the last 2000 years,
does not really represent our ordinary modes of reasoning,
but is rather a scheme by which we try to show other people
how our conclusions follow from our premisses. The arranging
of a string of syllogisms, as syllogisms, presents no serious
difficulty; from accepted premisses, a logical conclusion
follows, with almost mechanical precision. Professor Jevons
actually invented a logical machine, almost as simple as a
pennyintheslotmachine, which made clear that syllogistic
reasoning was at bottom mechanical. In reasoning the real
difficulty is concerned with the premisses that compose
the syllogism, not with the syllogism itself. Can we certify
that the premisses are true, and do we all agree about the
exact significance of the terms we use? This is the trouble,
and the only real trouble, involved in ordinary reksoning.
To follow out the sequence of syllogisms in a Euclidean
proposition is child's play.
Opposite is a scheme showing Euclid's chain of reasoning
for proving I, 32. The proposition and all the propositions
on which it depends are easy enough to follow up, and may
(with certain exceptions) be accepted. But when we come
to the axioms which form the ultimate premisses of the pro
positions, and examine them carefully, we begin to feel
doubt and difficulty. In short, we have reached the point
where the work of serious reasoning begins.
This is the first thing for boys to bear in mind: that
although mathematical demonstrations have every appearance
of being mere chains of syllogisms, we may not infer that
mathematical reasoning is deductive. The very contrary is
really the case. Mathematical reasoning is above all things
THE PHILOSOPHY OF MATHEMATICS
1
2
4
I
625
5
7
8
22
23
9
10
11
13
15
1(3
27
29
31
32
analytical, inductive; it cannot be reduced to the rules of
deductive logic. We dress the results up syllogistically
merely for exhibition purposes.
Impress upon the boys that in all reasoning beginners
tend to believe that the premisses are true if acceptable
consequences seem to follow from them, and the longer the
chain of apparently sound, intermediate links, the less sus
picious they become of any weakness in the first link; but
that conclusions are quite worthless unless the premisses on
which they first depend are unassailable.
It was John Stuart Mill who in the middle of the last
century first shook our faith in Aristotelian logic. Mill's own
position has since been shaken, but it was he who first gave
to the subject its proper outlook.
(E291) 41
626 CRAFTSMANSHIP IN MATHEMATICS
The " new " logic, the logic especially of the last thirty
years, insists upon this: that verbal explanations of meaning,
so long as they remain merely verbal, are futile. Merely
to " infer " one proposition from another, and to go on doing
this for ever, gives us nothing but unexplained " propositions "
at every step of the process. The old logic never even became
aware of the fatal confusion between assertion and sentence
that is covered by the word " proposition ". According
to it, if we get one proposition from another or others, we
have arrived at the end of a process of inference and have
obtained a " conclusion ". If " all men are feathered animals "
happened to be one of the premisses, the old logic did not
question that premiss at all, but proceeded to draw " con
clusions "; the truth or falsehood of the premisses was none
of its business. Its conclusions were thus often meaningless
and misleading; sham, not real. The new logic, on the
other hand, pays first attention to the premisses, knowing
that the 3ubsequent process of inference is a process relatively
elementary.
We must not allow our admiration of the Greeks to blind
us to their limitations and their failures. Greek mathematics
was great; modern mathematics is greater. In certain* regions
the Greeks failed where we feel they ought to have succeeded.
We feel, for example, that they ought to have anticipated
Descartes; and we feel that, with their power of generalization
and their love of philosophy, they ought to have given some
coherent account of the foundations of geometry. The
early pages of Euclid are definitely unsatisfactory.
Apparently the Greeks never realized that the foundations
of geometry were necessarily abstract. Even Euclid himself
could only look at space vaguely, and give some sort of popular
description of what he thought he saw there. And "to the
Greeks generally, geometry was always the " science of space"
of the physical space of experience in which we live. It is
quite impossible to base a coherent geometry on such a
foundation; the superstructure may be magnificent, but it
is always likely to overturn because of the instability of its
THE PHILOSOPHY OF MATHEMATICS 627
foundations. That the theorems of geometry are not affected
by earthquakes, that the Greeks could understand, but they
could not, or at least did not, understand that geometry has
nothing to do with physical space; that its " space " is its
own creation; and that it is simply the statement of the logical
relations between objects, defined by these relations alone.
Nonmathematicians still commonly suppose that the
early pages of Euclid, his " axioms " and " postulates ", are
profound and never to be questioned. There is, in point of
fact, not one of these axioms or postulates, the parallel
postulate alone excepted, which has anything but an historical
interest, or which embodies any permanent contribution to
science. Here the Greeks failed altogether.
Here is Mr. Bertrand Russell's opinion of Euclid.
" When Euclid is attacked for his verbosity or his obscurity
or his pedantry, it has been customary to defend him on
the ground that his logical excellence is transcendent, and
affords an invaluable training to the youthful powers of
reasoning. But at a close inspection, this claim vanishes.
His definitions do not always define, his axioms are not always
indemonstrable, his demonstrations require many axioms
of which he is quite unconscious.
" The first proposition assumes that the circles used in
the construction intersect an assumption not noticed by
Euclid because of his dangerous habit of using a figure. The
fourth proposition is a tissue of nonsense. Superposition
is a logically worthless device; for if our triangles are spatial,
there is a logical contradiction in the notion of using them;
if they are material, they cannot be perfectly rigid, and when
superposed they are certain to be slightly deformed from
the shape they had before. The sixth proposition requires
an axiom for proving that if D be in AB (the side of the
isosceles triangle ABC), and BD is < BC, the triangle, DBC
is < the triangle ABC. The seventh proposition is so thor
oughly fallacious that Euclid would have done better not
to attempt a proof. I, 8 involves the same fallacy as I, 4. In
I, 9 we require the equality of all right angles, which is not
628 CRAFTSMANSHIP IN MATHEMATICS
a true axiom since it is demonstrable. I, 12 involves the
assumption that a circle meets a line in two points or in none,
which has not in any way been demonstrated. I, 26 involves
the same fallacy as I, 4 and I, 8. Many more criticisms might
be passed on Euclid's methods, and on his conception of
geometry; but the above definite fallacies seem sufficient to
show that the value of his work as a masterpiece of logic
has been very grossly exaggerated."
Still, the Greeks did conceive the universe as a cosmos
subject to rule; they did recognize that the universe is, at
bottom, a mathematical affair. It was not until 2000 years
later that the foundations of their science were carefully
examined and found to be lacking in any sort of solidity
or permanence.
Axioms, Postulates, Definitions
Every conclusion rests on premisses. These premisses
are either selfevident and require no demonstration, or they
can be established only by demonstration from other proofs.
Since we cannot thus proceed in this latter fashion ad mfinitum,
geometry must be founded on a certain number of un
demonstrable propositions.
To these undemonstrable propositions, some of the
greatest mathematicians of the last fifty years have devoted
very serious attention. That there is still divergence of
opinion shows how difficult the subject really is. The several
books which are the separate or joint productions of Mr.
Russell and Professor Whitehead will be familiar to most
readers, and the books of equally eminent French and German
mathematicians will also be familiar to some. Such a mass
of authoritative literature will convince any nonexpert that
the subject is a very thorny one.
The standard of logical rigour in mathematics is now
greater than it has ever been. It is, however, still quite
permissible, in teaching, to make use of small boys' in
tuitions to use them, in some measure, as a reinforcing
THE PHILOSOPHY OF MATHEMATICS 629
basis when establishing elementary principles from data
derived from the boys' experience or from special experi
ment. For later serious work, geometrical intuitions are
not sufficiently trustworthy. One wellknown authority
describes intuitions as " a mere mass of unanalysed prejudice ".
Certainly boys' intuitions are necessarily often crude, often
meaningless, just shots in the dark. We shall return to the
question of intuition in the next chapter.
What are geometrical axioms? Kant said they were
synthetic a priori intuitions. But in that case they would
be imposed upon us with such a force that we could not
conceive their contraries, and this we now know we can do.
Are they, then, experimental truths? We do not make
experiments on ideal lines or ideal circles; we can make
them only as material objects. On what, therefore, would
experiment serving as a foundation for geometry be based?
The answer is simple. Since we constantly reason as if
geometrical figures behaved like solids, geometry can borrow
from experiments only the properties of those bodies. But
this involves an insurmountable difficulty, for if geometry
were an experimental science, it would not be an exact
science; it would be subjected to continual revision. Indeed
no rigorously invariable solid exists.
Thus the axioms are neither synthetic a priori intuitions
nor experimental facts; they are merely conventions.
Our choice amongst all possible conventions is guided
by experimental fact, but it remains free, and it is limited
only by the necessity of avoiding every contradiction. Thus
different geometries are possible according to our initial
choice of conventions. One of these geometries is not more
" true " than another (cf. the last chapter); it can only be
more convenient. Euclidean geometry is the most convenient
because (1) it is the simplest; (2) it sufficiently agrees with
the properties of natural solids, those bodies which we can
compare and measure by means of our senses.
Henrici (no mean geometer) called Euclid's definitions
" axioms in disguise ". Poincar6 (a much abler mathematician,
630 CRAFTSMANSHIP IN MATHEMATICS
perhaps the greatest of the last fifty years) called Euclid's
axioms " definitions in disguise ". As to Euclid's postulates,
some writers have seen in them merely statements which
limit the use of instruments in geometrical construction to
ruler and compasses, but this is a mere side issue, and has
nothing whatever to do with the nature of geometrical reason
ing.
Euclid himself grouped the 10th, llth, and 12th axioms
together with the three postulates, in one class, under the
name arn?/xara; the arrangement found in the modern editions
of his Elements is the work of his successors, the ground of
the alteration being, " the distinction between postulates and
axioms, which has become the familiar one, is that they
are the indemonstrable principles of construction and demon
stration, respectively ". This distinction is not accepted by
the modern school of geometers and should be ignored.
Views on the question are, however, discordant.
We referred in an earlier chapter to the desirability of
teaching boys carefully how to formulate their own de
finitions. The great purpose of a definition is the precise
discrimination, yes or no, of actual cases. The definition
must leave no doubt at all as to the identity of the thing
defined. With beginners there is bound to be superfluity
of statement, but refinement will come as the years go on.
Never provide pupils with definitions ready made.
I find that very few boys are able to grasp the significance
of the argument, pro and con, as to whether axioms (if, as
they seem to be, they are definitions in disguise) are a priori
intuitions, or experimental truths, or geometrical abstractions,
or whether they define conventions. The arguments are for
maturer minds. But it is easy to bring a boy to see that we
are all liable on occasions to admit as selfevident propositions
which we subsequently recognize not to be so.
Ask a class (a Fourth Form will do) to consider the axiom,
" two straight lines cannot enclose a space ". Accept their
usual definition of a straight line, viz. " the shortest distance
between two points ", as determined by a stretched string.
THE PHILOSOPHY OF MATHEMATICS 631
Then obtain from them the admission that the straightest
the most direct line (the geodesic) that can be drawn
between two points on the surface of a sphere may be de
termined in the same way; then show that this line is necessarily
part of a great circle. Next, 1 show that any two such lines
on the surface of a sphere, being parts of great circles, intersect
in two points; hence, at all events on the surface of a sphere,
any two " straight " lines must enclose a space.
But we live on the surface of the sphere, and any surface
we call plane, no matter how small, must be part of a spherical
surface, and therefore any two " straight " lines we draw,
being parts of great circles, must meet, at two points 180
apart. Even if the two lines we draw on the paper are what
we call " parallel ", the same thing applies.
Point out that this is not mere theory, it is prosaic fact,
for (1) the earth is known to be a sphere and therefore all
our socalled planes are parts of a spherical surface; and
(2) canal engineers actually have to allow 8" in the mile for
sphericity; if they made the bottom of the canal absolutely
" level' \ they would be cutting a channel which would eventu
ally emerge at the spherical surface. If 3 poles of equal length
be set up at equal intervals on a perfectly " level " stretch
of land, say along the straight six miles of the Bedford level
between Witney Bridge and Welsh's Dam, and a telescopic
sight be taken from the top of the first to the top of the
third, the line of sight will be seen to pass 5 or 6 feet below
the top of the second. If from one end of the Corinth canal
we look through the clear Greek air along the canal surface,
the earth's curvature is readily seen, and the amount of
curvature is easily measured by taking from one end of the
canal a tangential telescopic sight to meet some object, say
the hull of a ship, at the other end.*
In a Sixth Form the argument can be carried farther,
* There are no locks on the canal, and two of its four miles of length run
through a cutting with an average depth of 200 feet. Tested at any point of its
length, the canal surface is found to be perfectly "level", yet the water halfway
along is demonstrably several feet " higher " than at either end.
632 CRAFTSMANSHIP IN MATHEMATICS
and conviction carried home. At all events, enough can be
done (with other axioms as well as this) to make boys critical,
in the future, of principles that claim to be fundamental.
An elementary lesson on " infinity " may serve to lead up
to an analysis of such an axiom as " the whole is greater
than its part ". On two or three occasions I have known
useful Sixth Form discussions take place after the boys had
read the more elementary parts of Mr. B. Rus*sell's chapters
on Infinity: see, for instance, pp. 17982 of his Knowledge
of the External World. Some notion of the mathematical
significance of the term infinity should be given to Sixth
Form boys; but the difficulties are real. Mr. Russell is our
ablest expositor of these difficulties. His critical views are
drastically destructive; his constructive views are not accepted
by all mathematicians.
Mathematical Proof
We have all heard a small boy, when engaged in a lively
argument with another small boy, say suddenly, " You can't
prove it." What does he really want when he thus calls for
a certificate of proof? It is very hard to say. Verification by
demonstration? Authority?
Strictly, the proof of a proposition is its directly logical
syllogistic derivation from other propositions which we
know to be certain and necessary, and ultimately derivative,
therefore, from definitions and axioms. To that extent every
deduction from definitions and axioms is also the proof
of the conclusion reached by it.
When a distinction is made between proof and deduction,
the proof is regarded as the problem of deciding as to the
truth of an hypothesis, of confirming it or refuting it. The
proof of an affirmative is the refutation of the negative;
and vice versa.
The mathematical forms of statement that have been
devised to record the facts of a proof include the explicit
mention of all the considerations needed to justify it against
THE PHILOSOPHY OF MATHEMATICS 633
any attack. But this record is made after the proof has been
achieved, and in setting out the proofs we keep out of the
record all our unsuccessful attempts, all our " scrappaper "
work, and include just those few links in the main chain that
are well and truly forged.
Ultimately we go back to our definitions and axioms,
and it is here where we are really so susceptible to attack.
We never seem to be able to make ourselves armourproof.
Mr. Bertrand Russell, or somebody like him, will come
along and inevitably find a joint where he can inflict a nasty
wound. " Proof " is purely a question of degree. It is hope
less to attempt to find a means of dissolving mathematical
error completely, and of exhibiting Truth in a white light,
unassailable. The most we can do with boys is to train them
both to be always on the lookout for mathematical shams
and to hunt these down relentlessly.
It is fatuous to make Third or Fourth Form boys write
out the general " proof " of, say, the division process of
finding the algebraic H.C.F. What does the " proof " signify
to them? So it is with " proofs " all the way up the school.
When a Fifth Form boy has " proved " the binomial theorem
ask him what it is all about; closely crossexamine him and
show him how the term " prove " has been improperly used.
Let him establish the binomial theorem; let him show that
for a fractional index the theorem takes the same form as
for an integral index, but do not let him pretend to " prove "
either. Beware of using the term proof in connexion with
mathematical induction. Even the sometimes substituted
phrase " proof by repetition " is open to criticism.
44 Pure " Mathematics
If a strictly logical treatment of mathematics implies, as
some presentday mathematicians and others contend, a
strictly abstract treatment, the objects with which mathematics
deals are just symbols, devoid of content except such as is
implied in the assumptions concerning them. This abstract
634 CRAFTSMANSHIP IN MATHEMATICS
symbolism constitutes what is sometimes called " pure "
mathematics, everything else being, strictly, " applied "
mathematics, since it deals with concrete applications of an
abstract science. Thus all the ideas of pure mathematics
can (so it has been seriously contended) be defined in terms
which are not strictly mathematical at all, but are involved
in complicated thought of any description. If this be true,
all the propositions of mathematics might be deduced from
propositions of formal logic.
The distinction is not the same as the distinction of thirty
or forty years ago. Then, " pure " mathematics included all
ordinary work in algebra, geometry, and the calculus;/ applied
mathematics included such subjects as mechanics, surveying,
and astronomy. Pure mathematics was the mathematics of
the blackboard; applied mathematics was supposed to be
the mathematics of the laboratory, but too often experiment
played no part at all. The " pure " mathematician was a
very exclusive sort of person, rather despising those who
did the weighing and measuring necessary hacks they
admitted, but hacks all the same.
Nowadays " pure " mathematics tends to shrink into a
smaller compass. Formerly, the laboratory experiment with,
e.g., Fletcher's trolley came (as it still comes) within the
ambit of applied work, but all subsequent considerations of
the curve produced belonged to algebra and trigonometry,
and therefore constituted " pure " work. But not so now.
The curve itself is now recognized as a thing of ink or chalk,
and is therefore material; it is not really a geometrical curve,
it is a black thing or a white thing that we make, and is only
a crude representation of the true curve which, if we are
" pure " mathematicians, it behoves us to consider.
Thus " pure " mathematics tends to become a new*
subject, a subject in the border region between the mathematics
that ordinary people learn, and philosophy. The subject is
a very serious one, a subject within the realm of " pure "
or abstract thought; but it is not a subject for schools. It
was a prominent Church paper that, a few years ago, be
THE PHILOSOPHY OF MATHEMATICS 635
lauded the " pure " mathematician because he was a " good "
man; the " applied " mathematician was necessarily led by
his " impure " work to free thinking and infidelity! The
small circle of eminent philosophical mathematicians whom
we recognize as authorities on abstract mathematics must
be proud of their testimonial. But the great majority
teachers and all others who are engaged in ordinary worka
day mathematics, merely " applying " the basic principles
laid down by the few should ponder over the fate that
is said to await all infidels!
Is it not a little just a little absurd to pretend that
we teach " pure " mathematics in schools. The work we
do is all applied work, the different forms of which are all
a question of degree. As we go up the school, the concrete
work receives a gradually deepening tinge of abstractness,
but even in the Sixth the work is never more than partially
abstract. To claim that the Upper Form geometry we teach
is more " refined ", is " purer ", or is " more intellectual "
than mechanics or astronomy is merely to provoke ribaldry.
It is sometimes said that Newton was a " pure " mathe
matician. But was he? He spent his life in rounding off the
work of the astronomers from the time of the ancient Greeks
and Egyptians to the time of his predecessor Galileo. Even
the new mathematical weapon (the calculus) which he forged
was forged for the purpose of pushing ahead with his investiga
tions among real things. But he was certainly also a philosopher
if by this term we mean a thinker who looks to his foundations.
His researches did not, however, take him very far into
purely abstract mathematics. He was too busy with such
problems as that of showing that a falling stone and a falling
moon are subject to the same law.
Though men could not deny the tremendous success of
Newton's system of mechanics, though Laplace acclaimed it
as final, yet there remained questionings. One distinguished
critic after another felt doubts about his absolute space and
time, and Einstein, setting out to satisfy them, not only did
so, but in the doing he evolved a theory that included not
636 CRAFTSMANSHIP IN MATHEMATICS
only all that Newton had done but those other points which
Newton's theory could not be made to include.
Mathematics, like all other subjects, has now to take
its turn under the microscope and reveal to the world any
weaknesses there may be in its foundations. But this is not
work for schoolboys. To boys the main objects of mathe
matical study must continue to be real things, even if those
real things are only figures produced by ruler and compasses.
Mr. Russell criticized Euclid for drawing figures, because
those very figures were partly responsible for preventing
Euclid from building up the flawless system he aimed at.
Schoolboys, working on a lower plane, would, without figures,
be helpless.
Relative Values
In one respect at least, mathematics seems to be a subject
quite unlike other subjects. Its discoveries are permanent.
The theorem of Pythagoras for instance is as valid today as
2000 years ago. The majority of the mathematical truths
we now possess we owe to the intellectual toil of many
centuries, and a student who desires to become a mathe
matician must go over most of the old ground before he
can hope to embark on serious research. To the uninitiated
it is impossible to make mathematical truths clear. The
great theorems and the great results of mathematics cannot
be served up as a popular dish, and this inaccessibility of
the subject tends to make it rather odious to those whose
early grounding was of little account.
Although mathematics does not lead to results which are
absolutely certain, the results are incomparably more trust
worthy than those of any other branch of science. Still, if
mathematics stands aloof, if it is not turned to practical account
in other branches of science, it remains a useless accumulation
of capital, almost an accumulation of lumber.
Can it be maintained that, as an intellectdeveloping in
strument, mathematics ranks first amongst the different
subjects of instruction, or even " first among equals "?
THE PHILOSOPHY OF MATHEMATICS 637
Mathematical reasoning is in some respects simpler than
scientific reasoning; the data are clearer. It is simpler than
linguistic reasoning; there are no probabilities to weigh. It
is simpler than historical reasoning; there are no difficult
human factors to consider. But in one respect mathematical
reasoning is the most difficult of all, and this is because of
the inherent difficulties of mathematical analysis, whether the
analysis is the Third Form analysis of the data of a simple
geometrical rider, or whether it is the more serious analysis
in Sixth Form work. Relatively speaking, there is no other
difficulty. The analysis once effected, the rest is plain
sailing.
I attach very great value to mathematical instruction, but
I deny that the virtue of the instruction lies in anything of
the nature of supercertitude.
Picture making by Physicists: the Dangers
It has been said that, from the broad philosophical stand
point, the outstanding advances in the physics of the last
20 years have been the theory of Relativity, the theory of
Quanta, the theory of Wavemechanics, and the dissection
of the atom. But it would be more correct to say that it is
the surrender of our rather aggressive certitude about the
nature of things, and our recognition that we are still ignorant
of the nature of ultimate reality. We now know that our
pictures were all wrong. How faithful we were to the aether
as a quivering jelly of inconceivable density! How we loved
to tie knots in aetherial vortex rings!
Certain seaside resorts are on a cliff, with an upper
esplanade brilliantly illuminated at night, and, 10 or 15 feet
below, an unlighted walk protected by a low wall. The
shadows of passing people and vehicles on the upper esplanade
are cast upon this low wall, and may be watched by a person
seated on the lower level in the dark. Imagine such a person
to have been entirely cut off from human kind since his
early childhood, to be fastened to his seat permanently in
638 CRAFTSMANSHIP IN MATHEMATICS
the dark, and to see nothing but the shadows in front of him.
We may consider him endowed with powers of reasoning
and with some amount of mathematical power.
We can imagine him observing the shapes, sizes, move
ments, and velocities of the shadows, gradually sorting out
resemblances and differences, and eventually establishing a
number of equations embodying the whole of his sense
data. These equations would be strictly representative of
reality as he knew it. But suppose he now began to speculate,
and to attempt to infer from his equations the nature and
properties of the original things, animate and inanimate,
that had cast the shadows on the wall. Would not the rapidly
moving motorcar be given pride of place, and would not
the slowmoving human being be looked upon as of secondary
importance? Would not all his conjuredup mental pictures
unfailingly be a mere travesty of reality? In his allegory
of the cave, Plato warned us, more than 2000 years ago, of
fallacies of this kind.
Physicists are learning that the greater part of their
observations are not observations of reality but of the shadows
of reality. When they invent an atomic gymnasium and a
system of electronic gymnastics, they know well that they are
just speculating wildly. On the other hand they know that
the mathematical formula they have established is, though
uninterpretable, in some way representative of reality.
Einstein's formula for gravitation is universally accepted.
His cosmology is not accepted. No cosmology can be,
for it is necessarily hypothetical, speculative, fanciful.
Teach the boy that the physicist as a research worker
and mathematician is a man to be respected, but that while
we may admire his pretty pictures, we are quite certain that
none of these will ever make Old Masters.
Books to consult:
1. Professor Whitehead's books.
2. Mr. Bertrand Russell's books.
3. Mathematical Education, Carson.
THE PHILOSOPHY OF MATHEMATICS 639
4. Mysticism in Modern Mathematics, Hastings Berkeley.
5. Science and Hypothesis, Poincare.
6. Les Stapes de la philosophie mathdmatique , Brunschvicg
7. De la Certitude logique, Milhaud.
8 The Human Worth of Rigorous Thinking , Keyser.
CHAPTER XLVII
Native Genius and Trained Capacity
Russell versus Poincar6
Mathematical philosophers, like the philosophers of other
schools, naturally have greater faith in their own systems
than in the systems of their rivals. Over one point in particular
they are hopelessly at variance, namely, as to the respective
roles that logic and intuition play in the origin and development
of mathematical ideas.
It was Aristotle who worked out the principles of de
ductive logic, and his scheme was universally accepted
almost down to the close of the Victorian era. In the middle
of the last century, George Boole, a distinguished mathe
matician, pointed out how deductive logic might be com
pletely symbolized in algebraic fashion. Given any pro
positions involving any number of terms, Boole showed
how, by a purely symbolic treatment of the premisses, logical
conclusions might infallibly be drawn.
At the beginning of the present century (in 1901), Mr.
Bertrand Russell said,* " Pure mathematics was discovered
by Boole. His work was concerned with formal logic, and
this is the same thing as mathematics "; and again:) " The
fact that all mathematics is symbolic logic is one of the
greatest discoveries of our age, and the remainder of the
In the International Monthly. f Principles of Mathematics.
640 CRAFTSMANSHIP IN MATHEMATICS
principles of mathematics consists in the analysis of symbolic
logic itself. " In their Principia Mathematica> the aim of
Mr. Russell and Professor Whitehead is to deduce the whole
of mathematics from the undefined logical constants set forth
in the beginning. And in Signor Peano's Formulario, the
different branches of mathematics are " reduced to their
foundations and subsequent logical order ". Moreover, in
his work Les Principes des MatMmatiques, M. Couturat
expresses the opinion that the works of Russell and Peano
have definitely shown not only that there is no such thing
as an a priori synthetic judgment (i.e. a judgment that
cannot be demonstrated analytically or established experi
mentally), but also that mathematics is entirely reducible
to logic, and that intuition plays no part in it whatever.
But Henri Poincare, who was described by Mr. Russell
himself as " the most scientific man of his generation ",
flouted the logistic contention. He denied that logistic
(mathematical logic) gave any sort of proof of infallibility,
or that it is even mathematically fruitful. It did certainly
force us to say all that we commonly assume, and it forced
us to advance step by step. But its labels are labels of con
sistency and do not in any way refer to objective truth.
" The old logistic is dead." " True mathematics will continue
to develop according to its own principles." " Fundamentally
its development depends on intuition."
Mr. Russell, in reply, said that mathematical logic was
not " opposed to those quick flashes of insight in mathe
matical discovery " which Poincare " so admirably de
scribed ". Nevertheless, the main outlooks of the two men
seem to be radically opposed.
Mr. Russell has said elsewhere, " Mathematics is the
science in which we do not know whether the things we
talk about exist, nor whether the conclusions are true ".
Apparently, then, Mr. Russell admits at least that logistic
is not capable of discovering the mathematician's ultimate
premisses, and is therefore not capable of establishing the
truth of its final conclusions. It does, however, determine
NATIVE GENIUS AND TRAINED CAPACITY 641
the consistency of our conclusions with the premisses, and
this is its undeniable merit.
The Origin of New Mathematical Truths
If ultimate mathematical truth is not discoverable by
logistic, whence is its origin? Has it already an existence
(as some contend) independent of us personally, something
suprasensible, already complete in itself, existing from the
beginning of time, waiting to be discovered? is it of a priori
origin? or is it actually created by mathematicians?
The term a priori is ambiguous. Literally it signifies that
the knowledge to which it applies is derived from something
prior to it, i.e. is derivative, inferred, mediate. The metaphor
involved in " prior " suggests an infinite series of premisses.
But the term a priori is also often used to indicate that certain
general truths come to the mind, to begin with, as heaven
born conceptions of universal validity, and are thus " prior "
to all experience. Strictly, however, all a priori truths are
derived truths. But derived from what?
The mind seems to have a natural capacity for dictating
the forms in which its particular experimental data may be
combined. We may therefore correctly speak of the mind's
creative powers, though not of its innate ideas.
The mind's undoubted power of detecting identity and
difference, coexistence and succession, seems to be original
and inborn. Still, the power is exercised only on a contem
plation of actual things, from without or from within, and
all such primitive judgments are individual. The mind
compares two things and proclaims them to agree or disagree.
The judgment is immediate, and it is felt to be necessary;
it is irresistible and does not admit of doubt; it seems to be
independent and to hang upon nothing else, and seems
therefore to be primitive. But although the power is innate,
this does not mean that the judgments themselves are innate.
As primitive judgments are immediate, they are some
times described as intuitive.
(E291) 42
642 CRAFTSMANSHIP IN MATHEMATICS
Intuition and Reasoning
An intuition seems to be a general judgment immediately
pronounced concerning facts perceived. But an intuitive
judgment is as liable to error as is a reasoned judgment.
There is a natural tendency to ascribe to intuition a
peculiar authority, for it seems to confront us with an irre
sistible force foreign to the products of voluntary and
reflective experience. But knowledge derived from intuition
is as much experiential knowledge as directly conscious
knowledge, and it is just as fallible.
If we put on one side our purely primitive judgments,
it seems very probable that, fundamentally, intuition and
reasoning are identical, the former being instantaneous, the
latter involving the notion of succession or progress. The
difference then would be merely difference of time, every
judgment of the mind being preceded by a process of reason
ing, whether the individual is able to recollect it or not.
There are times when a great new truth suddenly comes
to the mind of a mathematician. The combination of factors
contributing to it seem to be a garnered knowledge derived
from accumulated experience, a complete analysis of the
given, a conscious connected reasoning, a systematic method
of working, a natural capacity, and, finally, a flash of intuition.
At some particular moment, the new truth flashes upon the
vision as if light from all the other contributing factors was
suddenly focused on the same point.
The Limitations of the Teacher's Work
Does not something of the same kind happen on a small
scale when an intelligent schoolboy is solving a difficult
problem? All ordinary methods of systematic attack may have
failed him, yet light suddenly comes. Whence? Who shall
say? Something from the rules of logistic, doubtless; some
thing from the boy's store of mathematical knowledge;
NATIVE GENIUS AND TRAINED CAPACITY 643
something from the boy's power of analysis of data; but
most of all from the boy's own native capacity. It has been
suggested that the truth suddenly emerges from a chance
combination of the boy's data. We may put it that way if
we like, but the " chance " seems to be very much more
than an affair of mere randomness.
If the boy's own native capacity is small, will the light
appear? Can skilful teaching make up for native deficiency?
In a considerable measure, yes; in a large measure, no.
I do not think that great mathematical skill can ever be
acquired by a boy with little natural mathematical endow
ment. We may meet with a considerable measure of sucfcess
when we teach mathematics to average boys, but we shall
never succeed in making such boys mathematicians. The
boy whose average mark is 60 per cent is probably greatly
indebted to his teacher. The boy whose average mark is
90 per cent probably owes very much more to nature.
Who can draw the line between the work of the boy's
mind and what is skilfully presented to his mind to work
upon? As teachers we have to teach methods, we have to
teach analysis, we have to teach logic, we have to provide
different types of mathematical knowledge. But can we do
more? Can we increase the boy's own native mathematical
capacity? Competent opinion answers that the old proverb
of the silk purse still holds good.
CHAPTER XLVIII
The Great Mathematicians of History
Few boys know anything about the great mathematicians
except by name. " No time," says the mathematical master,
a statement that cannot be denied. Yet, in order to include
644 CRAFTSMANSHIP IN MATHEMATICS
at least a little of the history of mathematics, I would sacrifice
something else. Let a few of the really great mathematicians
live once more, and let them be presented to the boys not
merely as mathematicians but as human personalities. If
it is to be but a few, who are the few to be? Personally I
would select, from the ancients, Archimedes and Pythagoras;
from Englishmen, Newton and Cayley; from Scotsmen,
Napier and Madaurin; from Irishmen, Hamilton; from
Frenchmen, Descartes and Pascal; from Germans, Gauss and
Leibniz. Others will plead for the inclusion of Sylvester,
Clerk Maxwell, Poincare, Einstein, and many more. Very
well: the more the better. Let those selected come back
and live their lives over again, and tell us exactly how they
came to make their great discoveries. Let the boys know
something of each great mathematical discovery, of each
mathematical leap forward, and of the men responsible.
One of the most readable books on the subject is Sir
Thomas Heath's A History of Greek Mathematics, but of
course it deals only with one period. It may be supplemented
by Mr. Rouse Ball's A Short Account of the History of Mathe
matics, D. E. Smith's History of Mathematics, Professor
TurnbulPs The Great Mathematicians, and F. Cajori's A
History of Mathematics.
CHAPTER XLIX
Mathematics for Girls
Opinions still differ about the relative mathematical
ability of boys and girls. My own conclusions, derived from
observations extending over a long period, are these, though
I express them with some reserve:
1. That a very small minority of girls in an average
MATHEMATICS FOR GIRLS 645
Form are as able, mathematically, as the ablest boys in the
corresponding Forms in boys' schools, perhaps 3 per cent
of girls against 10 per cent of boys.
2. That the average mathematical ability of a Form of
girls is rather lower than that of the corresponding Form of
boys.
3. That the interest of girls in mathematics is decidedly
less than that of boys.
4. That all girls should be compelled to take mathematics
for 2 years (say, 11 to 13), but at the end of this time girls
making no useful progress should give up the subject, arith
metic excepted, though the arithmetic should contain a
certain amount of quite informal algebra and geometry,
mainly by way of mensuration and the free use of formulae.
Thus, in a large school with three or four parallel Forms at
each stage from 13+ to 16+, the bottom Form would, as
a rule, be a nonmathematical Form.
However, the question is a woman's question, I would
even say a question for women teachers generally rather than
for mathematical mistresses only. Men are not likely to be
able to consider adequately all the factors involved, or,
indeed, to know them; and I am doubtful if a committee
consisting only of mathematical mistresses would find it
easy to free themselves entirely from the prejudice which
naturally attaches to one's own subject. A committee for
discussing the question should include a proportion of
nonmathematical mistresses, but only those who could
discuss the question objectively.
The Girls' School Committee of the Mathematical
Association issued a special Report on Mathematics in Girls'
Schools in 1916. The Report was revised in 1928, and
reissued in 1929. The Report is full of valuable suggestions,
and should be read by all who are engaged in teaching mathe
matics to girls. There are, however, a few very able mathe
matical mistresses whose attitude towards the Report is a
little critical.
64:6 CRAFTSMANSHIP IN MATHEMATICS
CHAPTER L
The School Mathematical Library
and Equipment
The Library
In the 134th number of ^^M^hermtical Gazett^l Dr.
W. P. Milne writes a particularly interesting and suggestive
article on this subject. A few paragraphs may be quoted:
" There is nothing more extraordinary in the educational
world at the present day than the change of attitude amongst
teachers towards the subject they have to teach. Teachers of
bygone ages were to a large extent content to regard their
pupils as so many buckets into which they were content
to pour a prescribed amount of intellectual material. In
most cases the teacher himself did not know the sources
whence came his stockintrade. Compare the attitude of
teacKers~bf the present day, which is consciously or uncon
sciously, explicitly or implicitly, utterly different. For the
teachers of today, their subject is not a finished structure;
it is an organic growth, ever growing, ever changing, ever
bein^ added to, ever having new methods devised and old
cast aside.
" Tr T do not think we quite realize what we owe to the
Technical Colleges in getting rid of the old view. For the
students of Technical Colleges, mathematics is essentially
an^ instrument. They are willing to take much upon trust.
They are afways on the lookout to use what they have
learned, as a weapon. The result is inevitable. Mathematics
acquired under these conditions may be imperfect, may be
roughandready, but the subject is at all events alive; it
is confined within no rigid barriers, and is eternally moving
and growing and changing. In the Technical College class
room, the propositions of geometry do not pass before the
MATHEMATICAL LIBRARY AND EQUIPMENT 647
audience in stiff and stately procession as the actors on the
Greek stage; they are rather as the people of the market
place, hot and throbbing with life. Another great and in
calculable influence in helping to overthrow the old rigid
and detailed view of a subject lies in the fact that men trained
on research lines are slowly percolating through the staffs
of the schools, and their attitude as teachers reacts on the
taught. The combined results of these movements and
tendencies is that both master and pupil recognize that it is
possible to know only a very little about this evergrowing
everchanging subject, but what knowledge they do possess
fills them with an eagerness to ask for more. There is in
fact a widespread feeling among teachers of mathematics
that every school ought to possess an uptodate mathematical
library, so that both teachers and taught may keep abreast
of the times in their school work, and also catch a glimpse
of the great regions of mathematical thought that lies
ahead.
" The school library should consist of two parts, one
for scholars and one for masters, though these two should
not be mutually exclusive. In general tone arid constitution
the masters' portion should be altogether heavier and more
serious than the scholars', because the teachers are older,
more experienced, and have greater width of knowledge and
intellectual power than the scholars. On the other hand,
the senior division of the library is not intended for highly
specialized experts in various branches of mathematics; the
books should present rather * First Courses ', as they are
popularly called, in the various subjects in which they deal.
What schoolmaster does not want to know something more
about that rapidly extending subject of Nomography? What
modern teacher of mathematics does not want to know
something about the theory of gunnery, submarines, stability
of aeroplanes? On the other hand, the scholars' library should
be more suited to^ their age and cajpacity; mathematical
recreations and puzzles;" Rouse Ball's Jlistory of Mathematics;
elementary history and theory of astronomy; the recent
648 CRAFTSMANSHIP IN MATHEMATICS
book on Astronomy by Professor R. A. Gregory; Pioneers
in Science, by Sir Oliver Lodge; and so forth.
" One cannot and ought not to lay down hard and fast
rules about the composition of the school library. A con
venient method, however, would be to collect and catalogue
the books under the headings of : (1) Biography, (2) History,
(3) Philosophy, (4) Mathematical Analysis, (5) Geometry,
(6) Applied Mathematics, (7) General. Such a composition
gives plenty of scope for variety and elasticity."
The whole article should be read by all mathematical
teachers.
In 1926, the Mathematical Association issued a.^ List of
oks^ " likely to prove useful for reference to teachers and
their more advanced scholars ". There are about JLOQ m a ^
mostly selected standard works, tried and proved, and re
commended for their real worth. No school can do better
than begin with these, and add others from time to time.
It should not be forgotten that the French and the Germans
are much keener mathematicians than we are, and that some
of their standard works in mathematics rank as classics.
Those unfamiliar with German may be interested to know
that enough about the language may easily be mastered in
a month for any technical German work to be read with ease.
(This does not of course apply to the speaking or writing of
German!)
At the end of the Report on Elementary Mathematics in
Girls' Schools (Mathematical Association) are some useful
appendices:
Appendix C. Lists of Books for the Libraries of Girls'
Schools (an admirable selection).
Appendix D. List of some of the Articles from the
Gazette that are of general interest to teachers. (Many others
may be found by referring to back numbers of the Gazette.)
Appendix E. List of Reports published by the Mathe
matical Association.
A bound copy of the Gazette should be added every year
to the teachers' section of the school library. The articles
MATHEMATICAL LIBRARY AND EQUIPMENT 649
are almost always helpful to the practical teacher, and every
copy of the Gazette is full of useful hints.
The teachers' section of the mathematical library should
also contain the Board of Education's Special Reports on
Educational Subjects, numbers 12, 13, 15, 16, 1834, all
dealing with mathematics in some form or other, all written
by wellknown experts.
Mathematical teachers may be reminded that, whilst
their school libraries are in the making (or at any other time),
they may, if they are members of the Mathematical Association,
borrow books from the Association's own library, which is
under the care of Professor Neville at Reading. A list of
books in this library is published by the Association.
Mathematical teachers should keep an eye on the columns
of the Mathematical Gazette, Nature, The Educational Outlook
and Educational Times, The Times Educational Supplement, and
The Journal of Education, for reviews of new books.
Equipment
Schools provided with a separate Mechanical Laboratory
are usually well stocked with apparatus for teaching mechanics
and practical mathematics, including such things as screw
gauges, calipers, verniers, spherometers, opisometers, slide
rules, a planimeter, cycloidal curve tracers, an ellipsograph,
linkmotion apparatus, a binomial cube, a variety of wooden
and wire geometrical models, and models to illustrate pro
jection. When there is a special mathematical laboratory,
the room is usually given up to the Lower and Middle Forms,
and few instruments of precision are to be found in it. Indeed
these are generally too expensive to be purchased, but they
can always be seen and examined at the Science Museum at
South Kensington.
Not all mathematical teachers are aware of the fine collection
of instruments in the mathematical section of the Science
Museum. I have often seen classes of boys under the
guidance of a teacher, examining the instruments, apparatus,
650 CRAFTSMANSHIP IN MATHEMATICS
and working models in the physics, mechanical, and engineer
ing sections of the Museum, but the mathematics and the
geodesy and surveying sections are usually deserted. Amongst
the things that all boys should see in these sections are the
following:
1. Instruments for drawing curves: trammels, ellipso
graphs, parabolographs, &c.
2. Ellipsoids, hyperboloids, paraboloids: various models
of curved surfaces formed by (i) intersecting layers of stiff
paper, (ii) series of stretched strings. (Some of these are
singularly attractive and wonderfully instructive. I remember
as a youth spending my very scanty supplies of pocketmoney
for several months in purchasing materials and tools for
reproducing some of these.) Pairs of intersecting cylinders
and of cones. Roof structures. Skew bridges.
3. Instruments for preparing perspective drawings from
plans and elevations.
4. Sundials and clocks
5. Calculating machines: Pascal's, Morland's, Stanhope's,
the Brunsviga.
6. Difference and analytical engines: Babbage's, Scheutz's.
7. Slide rules: straight, circular, spiral, cylindrical, and
gridiron types.
8. Instruments for solving equations.
9. Linear integrators, planimeters, integrometers, inte
graphs. Professor Boys' curve  drawing integrator, and
his model of a polar planimeter.
10. Harmonic analysers and integrators.
11. Jevons' logical machine.
12. Rangefinders and tacheometers.
13. Clinometers, azimuth compasses, prismatic com
passes, early theodolites, altazimuth theodolites, geodetic
theodolites, zenith sectors and telescopes.
14. Mine surveying and marine surveying instruments.
15. Air survey maps, geodetic triangulation maps, &c., &c.
The services of a wellinformed museum guide may be
MATHEMATICAL LIBRARY AND EQUIPMENT 651
obtained for the asking; he will accompany visitors on their
rounds, and explain the instruments under examination.
Many of the machinery models in the Science Museum
are in motion, being driven by compressed air. Visiting
boys should always be taken to see them.
Visits may also profitably be made to such places as
Cussons' Technical Works, Manchester; to Sir Howard
Grubb, Parsons, & Co.'s optical works at Newcastle; to
Hilger's; and to some of the betterknown instrument makers.
Teachers interested in practical measuring might read
Mr, F. H. Rolt's Gauges and Fine Measurements, the standard
work on measuring machines, instruments, and methods.
I am doubtful if any part of the mathematical equipment,
even instruments of historical interest, should be kept in the
school museum. The school museum is apt to be a holy
place, to be visited seldom, and then silently lest the dust
should be disturbed.
APPENDIX I
A QUESTIONNAIRE FOR YOUNG
MATHEMATICAL TEACHERS
1. In teaching arithmetic, to what extent is it advisable to give
a logical justification of the rules you teach at the time you teach
them? Illustrate your answer by reference to division of vulgar frac
tions, and to multiplication of decimals.
2. What is your general plan for teaching the tables to young
children? Doubtless you are convinced of the necessity for plenty of
dingdong work to make the tables perfect. What is the best means
of dealing with a visitor who suggests that such work is unintelligent?
3. Draft a few notes suitable for a Student Teacher, instructing
him how to proceed when teaching subtraction to beginners.
4. What is the proper function of socalled " mental " arithmetic?
Discuss the merits of (a) oral work, and (6) practice on paper, in
mathematical teaching, and assign to each type its proper place in
the different Forms of the school.
5. Where in the school would you begin algebra, and how? Out
line a suitable course of work for the first two years, and show how
you would link it up with the mensuration, arithmetic, and geometry.
6. A slowwitted boy of Form IV tells you that he does " not
quite see " how the product of V3 and V5 can be V 15. On question
ing him you find that he has failed to grasp the essentials of the lesson
you have just given. Sketch out a new and very simple lesson to meet
the case.
7. Would you include the theory of annuities in a school mathe
matical course? If so, on educational grounds or on utilitarian grounds?
8. What is your experience of teaching logarithms to bottom Sets?
Is the practice desirable? If so, on what grounds?
9. The pedagogical treatment of parallel lines is admittedly very
difficult, and no teacher would suggest that a rigorous treatment at
the outset is possible. How would you make the subject more and
more logically exacting in the successive Forms from II to VI?
653
654 CRAFTSMANSHIP IN MATHEMATICS
10. Defend the principle that formal definitions in geometry
should never be provided readymade by the teacher.
11. Criticize the value of the work on graphs commonly done in
schools. How would you modify such work?
12. In what part of the school mathematical course would you
first introduce the idea of incommensurables? Should it be included
at all in a school with a leaving age of 16? If so, justify your opinion.
13. Sketch out a course of numerical trigonometry for D Sets
(Forms III to V), introducing as much field work as possible and
reducing academic work to a minimum.
14. Whose business is it to teach mechanics, the teacher of mathe
matics or the teacher of physics? If the former, how is he to proceed
if he has had no training as an engineer? If the latter, how is the mathe
matical side of the subject to be dealt with effectively?
15. In a Sixth Form course of mechanics, how far is it (a) possible,
(b) desirable, to replace some of the elegant but rather academic and
useless problems in ordinary dynamics by an elementary course in
the dynamics of astronomy and geology?
16. Kepler 's Third Law is found to hold good for the earth as
well as for the other planets. How would you demonstrate to a Sixth
Form that this fact alone affords strong evidence that the earth itself
is a planet?
17. Do you find that school mathematics has improved since the
calculus has been taken up in Forms below the Sixth? Defend (a) its
inclusion in, (b) its exclusion from, such Forms. What do you con
sider to be the educational gain of a course of analysis in school work
for boys not going on to the University?
18. Mathematicians have investigated many geometrical curves
that have little relation to practical life. On the other hand, they
have almost neglected to investigate the curvature of such common
things as eggs. The reason sometimes put forward is that no two
exactly similar eggs have ever been discovered. Assuming this to be
a fact, does the fact justify the neglect? If it does, what have you to
say about the mathematics of biology generally?
19. The physiological exchange which is inseparable from active
life is conducted through limiting surfaces, external and internal.
Provided the form remains unchanged, the bulk of a growing cell,
tissue, or organ increases as the cube of the linear dimensions, the
surface only as the square. Accordingly, as growth proceeds, the
proportion of surface to bulk decreases, until a point of physiological
inefficiency is reached. Your biological colleague, who is not a
mathematician, appeals to you for help over this difficult question
of interrelation between growth and form, whether applied to ex*
ternal surfaces or to internal conducting tracts in plants and animals.
Sketch out a course of lessons dealing with the mathematics of this
biological " sizefactor ".
20. Sixth Form work on (a) Capillarity and (b) Viscosity is almost
APPENDIX 655
always treated mathematically in the main, experiment playing a
very small part. Assuming that this treatment is correct, it is clearly
the business of the mathematical master to be responsible for it.
Give an outline of three or four lessons in each subject, and say
exactly what help, and where, you would expect from your Physics
colleague.
21. A Sixth Form boy tells you his Physics teacher has said that
Van der Waals' equation is not sufficiently representative of the facts,
and he asks you if you will explain " the mathematical fallacy in the
equation ". Devise a suitable reply.
22. Write out brief notes of a lesson on Lissajous figures, includ
ing a demonstration of the method of establishing the general equations
 = sinraO, ^ sin(w6 + p), where m and n are proportional to the
a b
frequencies of the horizontal and vertical vibrations. Should the
mathematics or the physics master demonstrate the fact that the
gradual changes from one figure to another depend on the gradual
change of p? Why?
23. Have you found School Certificate examination requirements
clash with your own thoughtout plans for mathematical teaching?
If so, how? Are you quite sure that the clashing is real? If so, take a
recent School Certificate paper, modify it in accordance with your
own views, and see if your colleagues agree.
24. Here are two wellknown old problems, much too difficult for
the average boy:
(i) A spider at one corner of a semicircular pane of glass
gives chase to a fly moving along the curve before him, the fly
being 30 ahead when the chase begins. Each moves with its own
uniform speed. The spider catches the fly at the opposite corner.
(1) Trace the spider's path; (2) determine the ratio of the speeds
of the two insects.
(ii) A horse is tethered to a stake in the hedge of a circular
field, the rope being just long enough to enable him to graze over
half the area of the field. Determine the length of the rope in terms
of the diameter of the field.
Is it worth while spending time over such examples of the type of
(i) the first, (ii) the second? Discuss examination conundrums generally
and their injurious effect on rational school practice.
25. Young mathematical teachers generally try to adopt methods
which, in exposition, shall be mathematically flawless. Have you
ever known " intelligent " teaching to be productive of unintelligent
reception? If so, how do you explain it? Who is likely to be the more
to blame, the teacher or the boy?
26. What do you understand by a proof! Criticize the term as
commonly used in mathematical teaching. At what stage do you
advise the introduction of simple proofs? When would you introduce
656 CRAFTSMANSHIP IN MATHEMATICS
formal proofs of the fundamental theorems of geometry (congruent
triangles, angles at a point, parallels, &c.)? What is your opinion of
the value, to boys, of the usual book proofs of H.C.F. (algebra) and
the Binomial Theorem?
27. What do you consider to be the essential distinction between
" pure " and " applied " mathematics? A chalk or a plumbago line
drawn with the compasses is as concrete a thing as the iron rim to a
flywheel: are you therefore doing " purer " work in the classroom
than in the laboratory? How do you defend the use of the term
" pure " in any of your mathematical teaching except perhaps in the
work of those Sixth Form specialists who are capable of grasping the
A B C of mathematical philosophy?
28. How do you account for so few persons being interested in
mathematics, although the great majority of educated people must
have done a fair amount of mathematics at school? How would you
modify the present course of school mathematics in order to ensure
a greater permanent interest in the subject on the part of learners?
29. Plan out a course of work for the Sixth Form, specialists and
nonspecialists alike, calculated to develop and maintain a lifelong
and intelligent interest in mathematics, the course not to be so recre
ational that rigour is seriously sacrificed.
30. What commonly taught topics would you advise should be
deleted from a school course? Why? What others would you substi
tute?
31. A boy asks the question, " Exactly when did the twentieth
century begin?'* How would you answer it? How would you deal
with his meridian difficulty, especially if he lived on the 180 meridian?
How would you make this difficulty a good jumpingofT place for
introducing first notions of Relativity?
32. Devise courses of instruction for:
(i) Junior Elementary Schools, 8+ to 11 + .
(ii) Senior Elementary Schools, 11+ to 14f.
(iii) Central Elementary Schools, 11+ to 15 + .
(iv) Preparatory Schools, 8+ to 13 + .
(v) Secondary Schools, 11+ to 18 + .
In the last case assume that the five Forms Upper III, Lower and
Upper IV, Lower and Upper V, each consists of four graded Sets,
A, B, C, and D. Show how the work of the Sets should differ in (i)
content, (ii) ^treatment.
33. Your school is of sufficient importance to receive each year
from a leading Training College a few students who have taken good
degrees in mathematics, and you are entrusted with the purely prac
tical side of their professional training. What measures would you
adopt to make such training effective?
34. It is sometimes said that a classical training is " necessarily "
productive of a much greater appreciation of the beautiful than is a
mathematical training; that a real appreciation of a beautiful thing
APPENDIX
657
is always accompanied by a glow, whereas a mathematician's apprecia
tion is always reasoned and therefore cold. The contention is, of
course, silly, but how would you formulate a defence that would
carry conviction to an opponents mind?
35. You are appointed chief mathematical master of a new school
where a room 36 ft. by 25 ft. has been set aside for a mathematical
laboratory. Give full details of the fittings and equipment you would
provide for elementary and for advanced work, the expenditure to be
limited to 500.
36. What is the proper function of school textbooks in mathe
matics, apart from the exercises they provide? Distinguish between
textbooks for Lower Forms and those for Higher.
APPENDIX II
NOTE ON AXES NOTATION
In drawing graphs I have followed the usual notation. But the
following figures indicate a useful reform:
Fig. i
1 . Mark the x and y axes merely by arrows pointing in the positive
direction. (Figs. 1 and 2.)
( K 291 ) 43 a
658 CRAFTSMANSHIP IN MATHEMATICS
2. Indicate dimensions by means of singleheaded lines instead of
the draughtsman's usual doubleheaded lines, the direction of the
arrow indicating the actual direc
tion in which the line is to be
measured. For instance, the
length QR is +(x + 1J), and the
length RP is \(y 1); the latter
is the difference in the dimen
sions measured in the positive
direction. (Fig. 1.)
3. Do not use a dimensional
line for length r, since by con
vention a radius vector is always
taken with the positive sign.
(Fig. 1.)
4. For angles, use a singleheaded (not a doubleheaded) arrow,
and let the arrowhead indicate the direction of rotation. (Fig. 2.)
O
INDEX
Acceleration, 428.
Accuracy, 18, 28.
Adams, 20.
Addition. 23.
Advanced mathematics, 553.
Algebra, addition and subtraction, 127.
and arithmetic in parallel, 132.
beginnings, 122.
cyclic expressions, 202.
formal beginnings, 125.
geometrical illustrations, 134.
links with arithmetic and geometry,
132.
multiplication, 130.
product distribution, 190.
2 notation, 193.
Algebraic equations, 205.
roots of, 205.
roots of biquadratics, 208.
roots of cubics, 208.
roots of quadratics, " either . . .
or ", 205.
expansions, 195.
expressions, complex forms, 194.
factors, 177.
geometrical models for, 179.
identities to be learnt, 204.
manipulation, 177.
phraseology, 181.
signs, 125.
symmetry, 201.
Amicable numbers, 597.
Andrews, 607.
Analysis of algebraic problems, 215.
of curves, 465.
of geometrical riders, 319.
Animal locomotion, 585.
Apothecaries' weight, 49.
Applied mathematics, 489, 634.
Approximation methods, 412.
A priori truths, 641.
Archimedes, 480, 486, 644.
spiral, 571.
Areas and volumes, 50.
Aristotle, 625, 639.
Arithmetic, 23.
commercial, 109, 117.
( E 291 )
Arnoux, 607.
Art, proportion and symmetry in, 558,
58